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LINEAR ALGEBRAChapter 1- Linear Equations in Linear Algebra
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Topics of StudyChapter 1 Linear Equations in Linear Algebra Chapter 4 Vector Spaces1.1 Systems of Linear Equations 4.1 Vector Spaces and Subspaces 1.2 Row Reduction and Echelon Forms 4.2 Null Spaces, Column Spaces, and Linear Transformations1.3 Vector Equations 4.3 Linearly Independent Sets; Bases1.4 The Matrix Equation Ax = b 4.4 Coordinate Systems1.5 Solution Sets of Linear Systems 4.5 The Dimension of a Vector Space1.6 Applications of Linear Systems* 4.6 Rank1.7 Linear Independence 4.7 Change of Basis1.8 Introduction to Linear Transformations 4.8 Applications to Difference Equations*1.9 The Matrix of a Linear Transformation 4.9 Applications to Markov Chains*1.10 Linear Models in Business, Science, and Engineering
Chapter 5 Eigenvalues and EigenvectorsChapter 2 Matrix Algebra 5.1 Eigenvectors and Eigenvalues2.1 Matrix Operations 5.2 The Characteristic Equation2.2 The Inverse of a Matrix 5.3 Diagonalization2.3 Characterizations of Invertible Matrices 5.4 Eigenvectors and Linear Transformations2.5 Matrix Factorizations 5.5 Complex Eigenvalues2.6 The Leontief Input-Output Model* 5.6 Discrete Dynamical Systems* 5.7 Applications to Differential Equations*Chapter 3 Determinants 3.1 Introduction of Determinants Chapter 6 Orthogonality and Least Squares3.2 Properties of Determinants3.3 Cramer’s Rule, Volume, and Linear Transformations
Linear Algebra Name:____________________________________2
Lesson- Systems GraphicallyDate:_____________________________________
Objectives: solve systems of equations graphically determine whether a system of linear equations is consistent and independent, consistent and
dependent, or inconsistent
Classifying Systems of Equations:
Consistent System: Inconsistent System:
Independent: Dependent:
Without graphing, state whether each of the following systems are consistent and independent, consistent and dependent, or inconsistent. Explain your reasoning.
2y + 3x = 64y = 16 – 6x
(1) 40y – 35x = 557x = 8y – 11
Solve the following systems of equations graphically:
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y
x
y
x
y
x
(2) y = 3x – 2 x + y = 6
Solve the following systems of equations algebraically:(3) x + 4y = 26
x – 5y = -10
(4) 2x + 3y = 125x – 2y = 11
(5) -3x + 5y = 126x – 10y = -21
Linear Algebra Name:____________________________________Lesson- Linear Systems Applications- Graphical
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Date:_____________________________________
Objective: solve real-world applications graphically with systems of equations
(1) Wayne started with 50 gallons of water in his pool, and he is filling it at a rate of 10 gallons per minute. His next-door neighbor Ted started with 20 gallons of water in his pool, and he is filling it at a rate of 15 gallons per minute. Which system of equations could you use to find when the pools will contain the same amount of water?
(A) y = 50 + 15xy = 20 + 10x
(B) y = 50 + 10xy = 20 + 15x
(C) y = 50 – 15x y = 20 – 10x
(D) y = 50 – 10x y = 20 – 15x
(2) Cramer’s Café is a local eatery that plans on selling copies of their famous recipes in a cookbook to raise funds for renovations. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will be sold for $6 each.
(a) Write equations that represent the cost and income from these cookbooks.
(b) Graph the equations.
(c) Use your graph from part (b) to determine how many cookbooks must be sold before Cramer’s Café makes a profit.
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y
x
(3) Teknotronics, Inc. needs to buy new software for their office computers. The Premium package costs $13,000 plus $500 for each additional site license. The Deluxe package costs $2,500 plus $1,200 for each additional site license.
(a) Write equations that represent the cost of each software package.
(b) Graph the equations.
(c) Use your graph from part (b) to estimate the break-even point of the software costs.
(d) If Teknotronics, Inc. plans to buy 10 additional site licenses, which software will cost less?
(4) Megan is thinking about leasing a car for two years. The dealership says that they will lease her the car she has chosen for $326 per month with only $200 down. However, if she pays $1600 down, the lease payment drops to $226 per month.
(a) Write equations that represent the amount Megan will pay with each plan.
(b) Graph the equations.
(c) Use your graph from part (b) to determine the break-even point when comparing these lease options.
(d) Which 2-year lease should she choose if the down payment is not a problem? Explain your answer.
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y
x
y
x
(5) A tub containing 16 gallons of water is draining at a rate of 1 gallon per hour. A basin of 3.5 gallons of water is draining at a rate of 1 gallon every 6 hours.
(a) Write a system of equations that represent y, the number of gallons left in the container after x hours.
(b) Graph the equations.
(c) If both containers began draining at the same time, how soon will the tub and basin hold the same amount of water?
(d) When the amounts are equal, how much water will be in each container?
(6) Paloma has $1500 in a savings account. She adds $30 to her account each month. Jose has $2400 in his savings account. He withdraws $30 from his account each month.
(a) Write a system of equations that represents this situation.
(b) Graph the equations.
(c) In how many months will they have the same balance in their savings accounts?
(d) What will be the balance in each account?
Linear Algebra Name:____________________________________7
y
x
y
x
Lesson- Linear Systems Applications- AlgebraicDate:_____________________________________
Objectives: solve real-world applications algebraically with systems of equations
(1) In January, Paloma’s long-distance bill was $5.50 for 25 minutes of calls. In February, her bill was $6.54 for 38 minutes of calls. Find the flat rate and charge per minute for long distance that the phone company is charging Paloma.
(2) Last year, the high school rugby team bought team shirts for $17 each and socks for $5. The total purchase was $315. This year, they bought the same number of items, but the total purchase was $342. This was because the shirts were now being sold for $18 and the socks for $6.
(a) Write a system of equations that represents the number of shirts and socks bought each year.
(b) Using your answer from part (a), algebraically find the number of shirts and socks the team bought each year.
(3) A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered.
(a) Write a system of equations that describes the above situation.
(b) Using your answer from part (a), find the total number of adults and the total number of children in the group.
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(4) A mail order company charges for postage and handling according to the weight of the package. A light package is charged $2 and a heavy package is charged $3. An order of 12 packages had a total postage and handling charge of $29. Write and solve a system of equations to determine how many light packages and how many heavy packages were in the order.
(5) The Polynomial Park’s Recreation Department ordered a total of 100 baseballs and bats for the summer baseball camp. Each baseball costs $4.50, and the bats cost $20.00 each. The total purchase price cost $822. Write and solve a system of equations to find how many of each item was ordered.
(6) Mr. McAlley is writing a test for his history class. The test will have true and false questions worth two points each and multiple choice questions worth four points each, for a total of 100 points. He wants to have twice as many multiple choice questions as true and false questions.
(a) Write a system of equations that represents the number of each type of question.
(b) Use your answer from part (a) to find how many of each type of question will be on the test.
(c) If most of his students can answer true and false questions within 1 minute and multiple choice questions within 1½ minutes, will they have enough time to finish the test in 45 minutes?
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(7) Tyrone is responsible for checking a shipment of technology equipment that contains laser printers that cost $700 each and color monitors that cost $200 each. He counts 30 boxes on the loading dock. The invoice states that the order totals $15,000.
(a) Write a system of two equations that represents the number of each item.
(b) Use your answer from part (a) to find how many laser printers and how many color monitors were delivered.
(8) Megan exercises every morning for 40 minutes. She does a combination of step aerobics, which burns about 11 Calories per minute, and stretching, which burns about 4 Calories per minute. Her goal is to burn 335 Calories during her routine.
(a) Write a system of equations that represents Megan’s morning workout.
(b) Use your answer from part (a) to find how long she should participate in each activity in order to burn 335 Calories.
Linear Algebra Name:____________________________________Lesson- Systems of Linear Equations- Matrices (1.1)
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Date:_____________________________________
Objectives: solve a system of linear equations using elementary row operations.
Definition of a Linear Equation in n Variables:
A linear equation in n variable has the form ,
where the coefficients are real numbers (usually known). The number of is the leading coefficient and is the leading variable.
In this section, we are interested in the collection of several linear equations. The collection of these linear equations are referred to as a system of linear equations.
Definition of System of m Linear Equation in n Variables:A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables:
where are constants.
Example:
Consider the following system of linear equations:
In matrix form it would look like this:
Gaussian Elimination
Matrices can be used to represent systems of equations, which we then try to solve. We would do this by manipulating the matrix into a simpler form. We can do this using some Elementary Matrix Operations.
There are three kinds of elementary matrix operations.1. Interchange two rows.2. Multiply each element in a row by a non-zero number.
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3. Multiply a row by a non-zero number and add the result to another row
Example 1:
Solve the following system using Gaussian elimination.1x + 2y = 13x + 7y = 2
Solution:
The goal is to reduce the matrix to something easy to work with, namely something with all zeros below the diagonal.
What did I do?
Using the reduced form we can now determine that the solution is as follows:
Example 2:
In matrix form:
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Solution:
Exercises:
1. 3x – 2y = 12x – 3y = 9
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2. 3x + 2y = 54x + 5y = 7
3.
HW- read 1.1 do p10 #1-21 every other oddLinear Algebra Name:____________________________________Lesson- Row Reduction and Echelon Forms (1.2)
Date:_____________________________________
Objectives: solve a system of linear equations using elementary row operations.
Special Matrices:1. A square matrix is a matrix with the same number of rows as columns. 2. A diagonal matrix is a square matrix whose entries off the main diagonal are zero. 3. An upper triangular matrix is a matrix having all the entries below the main diagonal equal to zero. 4. A lower triangular matrix is a matrix having the entries above the main diagonal equal to zero.
Row Reduction and Echelon Forms (rref)
DEFINITION 1: A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry.
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DEFINITION 2: A leading entry of a row is the leftmost nonzero entry in a nonzero row. DEFINITION 3: A rectangular matrix is in echelon form if it has the following properties:
(1) All nonzero rows are above any rows of all zeros. (2) Each leading entry is in a column to the right of the leading entry of the row above it. (3) All entries in a column below the leading entry are all zeros.
DEFINITION 4: If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form:
(4) The leading entry in each nonzero row is 1 (5) Each leading 1 is the only nonzero entry in its column.
Note: The instance of using the rref to solve a system is known as Gauss-Jordan Elimination.
Examples:
and are matrices in reduced row echelon form.
is not in reduced row echelon form but in row echelon form since the matrix has the first 3 properties and all the other entries above the leading 1 in the third column are not 0.
is not in row echelon form (also not in reduced row echelon form) since the leading 1 in the second row is not in the left of the leading 1 in the third row and all the other entries above the leading 1 in the third column are not 0.
Example:
Transform matrix A into rref:
Solve the following linear system by finding the rref equivalent of the corresponding matrix.
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Important result: Every nonzero matrix can be transformed to a unique matrix in reduced row echelon form via
elementary row operations. If the augmented matrix can be transformed to the matrix in reduced row echelon form
via elementary row operations, then the solutions for the linear system corresponding to is exactly the same as the one corresponding to .
HW- read 1.2 do p21 #1-29 every other oddLinear Algebra Name:____________________________________Lesson- Vector Equations (1.3)
Date:_____________________________________Properties of Vectors in R n
Definition of vector:
An n-vector in is an matrix: (also known as a column vector)
where are the components of u.
If n = 2, we have what is known as an (r-two) vector. That is, a vector containing only two entries.
Examples:
Definition of matrix operations:
Let and
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The sum of the vectors u and v is the vector is the vector: Note: .
Let c is a scalar, then the scalar multiplication of a vector is:
The scalar multiplication rule allows us to write the linear system: as
Example:
Given: Find 3u, u + v, 2u – 3w
Graphical representation of
Example:
Given
Graphically find u + v
Example:
Given
Graphically find 2(u + v)
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Solution: Solution:
Graphical representation of
General structure:
Example:
Given
Graphically find u + v
Solution:
Example:
Given
Graphically find 2(u + v)
Solution:
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Properties of vector addition and scalar multiplication:Let u, v, and w be vectors in , and let c and d be scalars. Then,
1.2.3.4.5.6.7.8.
Note- A vector whose entries are all 0 represents a zero vectorLinear Combinations
Definition of linear combination:
If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors.
AKA- A linear combination is the vectors and scalars that, when added, gives a particular vector.
For example, suppose a = 2b + 3c, as shown below.
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= 2 1
2 + 3
3
4 =
2*1 + 3*3
2*2 + 3*4
a b c
Note that 2b is a scalar multiple and 3c is a scalar multiple. So, a is a linear combination of b and c.
The more textbook-y version:
Let vectors in a real vector space . A vector y in is called a linear combination of with weights of if .
Example:
Let
.
Since
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,
is a linear combination of because it can be (and is) a sum of scalar multiples.
Example:
Let
be vectors in the vector space consisting of all matrices. Then,
.
That is, is a linear combination of To verify a linear combination:
1. Set up the augmented matrix and row reduce it.2. Use the reduced form of the matrix to determine if the augmented matrix represents a consistent system of
equations. If so, then is a linear combination of the others. Otherwise, it is not.
Example:
Let , , and .
Determine whether b can be written as a linear combination of and .
Example:
Is the vector a linear combination of the vectors below?
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Example:
Is the vector a linear combination of the vectors
.
To write a vector as a linear combination of other vectors:
1. Set up the augmented matrix and row reduce it.2. Use the reduced form of the matrix to determine if the augmented matrix represents a consistent system of
equations. If so, then is a linear combination of the others. Otherwise, it is not.3. Find a solution (or set of solutions)
Example:Write the vector as a linear combination of the vectors: , ,
and
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Example:Given below, write w, x, y, and z as linear combinations of u and v.
Example:Mike has a test average of 75, a quiz average of 60, a homework average of 80, and a final exam score of 90. Determine his final average if tests are weighted at 0.4, quizzes at 0.3, homework at 0.1, and the final exam at 0.2.
Span {v} and Span {u, v}
Span {v} represents the set of all possible scalar multiples of a given vector v passing through v and 0.
Span {u, v} represents the plane containing u, v, and 0. It is a two-dimensional figure generated by the intersection of u and v. Note- v cannot be a multiple of u otherwise u and v would overlap.
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Example:
Let
.
Does ? (Hint- let and prove the existence of real values a,b, and c.)
Example:
List 5 vectors in For each vector, show the weights on used to generate the vector and list the 3 entries of the vector.
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HW- read 1.3 do p32 #1-29 every other oddLinear Algebra Name:____________________________________Lesson- The Matrix Equation Ax=b (1.4)
Date:_____________________________________
Objective: To learn the properties associated with the product of a matrix and a vector.
Meaning: Each entry in matrix A gets multiplied by the corresponding entry in vector x and the results are added (foundation of matrix multiplication).
Example:
Example:
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Example:
Let and . Is u in the subset of spanned by the columns of A? Why or why not?
Row-Vector Rule for Computing Ax
If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row I of A and from the vector x.
Compute the products using the row vector rule for computing Ax. If a product is undefined, explain why.
1)
2)
If A is an , u and v are vectors in , and c is a scalar, then:a. A(u + v) = Au + Avb. A(cu) = c(Au)
Example25
Prove A(u + v) = Au + Av using , and u, v in
ExampleProve A(cu) = c(Au) using A(cu) as a linear combination of the columns of A using the entries in cu as weights in .
Example:
Let , , and . It can be shown that 2u – 3v – w = 0. Use this fact, and no row operations) to
find and that satisfy the equation:
Example:Rewrite the matrix equation below in symbolic form as a vector equation, using symbols for the vectors and for scalars. Define what each symbol represents, using the data given in the matrix equation.
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HW- read 1.4 do p40 #1-29 every other odd, 37, 37Linear Algebra Name:____________________________________Lesson- Solution Sets of Linear Systems (1.5)
Date:_____________________________________
Objective: To learn how to use vector notation to give explicit and geometric descriptions of solution sets of linear systems
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Parametric Vector Form: written in the form of:
Example 1: Write in parametric form.
Homogeneous Linear Systems
Definition: A homogeneous system of linear equations is one in which all the numbers on the right hand side are equal to 0.
The homogenous system Ax = 0 always has the solution x = 0 (trivial solution). It follows that any homogeneous system of equations is consistent.
Any non-zero solutions to Ax = 0, if they exist, are called non-trivial solutions. These may or may not exist.
Example 2: Solve Ax = 0 if but will accomplish the same thing.
Can also be written as:
Example 3: Solve Bx = 0 if
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Example 4: Solve Cx = 0 if
Example 5: Solve Dx = 0 if
Example 6: Solve Ex = 0 if
Non-Homogeneous Linear Systems
Definition: A non-homogeneous system of linear equations is one in which all the numbers on the right hand side are not equal to 0. They are instead equal to some non-zero vector b.
Solution to a non-homogeneous linear system: If is consistent for some given b and p is a solution, then the solution set of is the set of all vectors of the form where is any solution of the homogeneous equation Ax = 0.
Example 7: Solve Fx = G if:
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Recap:
HW- read 1.5 do p47 #1-23 oddLinear Algebra Name:____________________________________Lesson- Applications of Linear Systems (1.6)
Date:_____________________________________
1. Jesse, Maria and Charles went to the local craft store to purchase supplies for making decorations for the upcoming dance at the high school. Jesse purchased three sheets of craft paper, four boxes of markers and five glue sticks. His bill, before taxes was $24.40. Maria spent $30.40 when she bought six sheets of craft paper, five boxes of markers and two glue sticks. Charles, purchases totaled $13.40 when he bought three sheets of craft paper, two boxes of markers and one glue stick. Determine the unit cost of each item.
Let p represent the number of sheets of craft paper.Let m represent the number of boxes of markers.Let g represent the number of glue sticks.
Express the problem as a system of linear equations and solve to determine the unit cost of each item.
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2. Rafael, an exchange student from Brazil, made phone calls within Canada, to the United States, and to Brazil. The rates per minute for these calls vary for the different countries. Use the information in the following table to determine the rates.
Month Time within Canada (min)
Time to the U.S. (min)
Time to Brazil(min)
Charges($)
September 90 120 180 $252.00October 70 100 120 $184.00November 50 110 150 $206.00
Let c represent the rate for calls within Canada.Let u represent the rate for calls to the United States. Let b represent the rate for calls to Brazil.
3. Calculate the number of minutes that Carlos called within Canada, to the United States, and to Mexico during the month of December. The charges are 28¢/min within Canada, 30¢/min to the U.S., and 84¢/min to Mexico if the following conditions applied:
His total bill for the month was $90.96 He talked twice as long to Mexico as he did to the U.S. The total number of minutes spent talking within Canada and to Mexico was 122.
Let represent the number of minutes within CanadaLet represent the number of minutes to the United StatesLet represent the number of minutes to Mexico
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4. Tracy, Danielle and Sherri bought snacks for a girls’ sleepover. They each bought the items shown in the following table at the local convenience store:
Number of bags of potato chips
Number of litres of pop
Number of chocolate bars
Cost($)
4 4 6 21.003 2 10 20.882 3 4 13.17
Calculate the unit price of each snack purchased by the girls.
5. A plant engineer wishes to calibrate the flow meters in the Distillate and Underflow of a distillation column that is used to purify methanol in a mixture of alcohols. He asks a chemical technologist for assistance. The engineer sets the column Feed rate at 100 lb/min. The technologist obtained samples from the Feed, Distillate and Underflow of the distillation column and analyzed them by Gas Chromatography to determined the concentrations shown in the table below. What are the flow rates of distillate and underflow? Solve this using matrices.
Feed Distillate Underflow
Methanol (%w) 50.0 96.0 38.5
Ethanol (%w) 30.0 2.0 37.0
1-Propanol (%w) 20.0 2.0 24.5
Answer: 20.0 lb/hr Distillate and 80.0 lb/hr Underflow.
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6. Aluminum oxide and carbon react to create elemental aluminum and carbon dioxide:
For each compound, construct a vector that lists the number of atoms of aluminum, oxygen, and carbon.
7.
HW- read 1.6 do p54 #1-15 oddLinear Algebra Name:____________________________________Lesson- Linear Independence (1.7)
Date:_____________________________________
Definition of linear dependence and linear independence:
An indexed set of vectors in a vector space is said to be linearly independent if the vector equation
has only the trivial solution.
The vectors kvvv ,,, 21 in a vector space R are said to linearly dependent if there exist constants, , not all 0, such that .
Linear Independence of Matrix Columns
The columns of matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution.
Sets of One or Two Vectors
A set of two vectors is linearly independent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other.
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Sets of Two or More Vectors
The nonzero vectors in a vector space V are linearly dependent if and only if one of the vectors , is a linear combination of the preceding vectors .
If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set in is linearly dependent if p > n.
Procedure to determine if are linearly dependent or linearly independent:
1. Form equation , which lead to a homogeneous system.2. If the homogeneous system has only the trivial solution, then the given vectors are linearly independent; if it
has a nontrivial solution, then the vectors are linearly dependent.
Example 1:
. Are and linearly independent?
Solution:
Therefore, and are linearly independent.Example 2:
Are and linearly independent?
Solution:
Therefore, and are linearly dependent
Example 3:
Determine whether the following set of vectors in the vector space consisting of all matrices is linearly independent or linearly dependent.
.
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Solution:
0000
0201
1203
1012
321332211 cccvcvcvc
Giving,
.
The homogeneous system is:
The associated homogeneous system has only the trivial solution: .
Therefore, and are linearly independent.
Example 4:
Determine if the vectors are linearly independent. Justify.
Example 5:
a. For what values of h is in Span ?
b. For what values of h is linearly dependent?
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Example 6:
Determine whether the following are linearly dependent.
a.
b.
HW- read 1.7 do p60 #1-29 every other oddLinear Algebra Name:____________________________________Lesson- Intro to Linear Transformations (1.8)
Date:_____________________________________
Objective: To learn how to apply linear transformations in a vector space.
A function F from the set V to the set W is a rule that assigns to each element x in V exactly one element F(x) in W. (AKA- definition of a function as you might see in college algebra with trig.)
A transformation (or function or mapping) T from to is a rule that assigns to each vector x in a vector T(x) in . The set is called the domain of T and is called the codomain of T. This is denoted as: . For x in , T(x) in is known as the image of x under T. The set of all possible images T(x) is called the range of T.
Example:
Let , , , ,
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and define a transformation T: by T(x) = Ax so that: .
a. Find T(u), the image u under the transformation T.b. Find an x in whose image under T is b.c. Is there more than one x whose image under T is b?d. Determine if c is in the range of the transformation of T.
Example:
Let , , define a transformation T: by T(x) = Ax
and define a transformation T: by T(x) = Ax so that: .
a. Plot vectors u and v and generate span(u,v)b. Conduct the transformations Au and Av and plot the resulting vectors and span(Au, Av).
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Definition: A transformation is linear provided that:
(a) T( u + v) = T(u) + T(v), and (b) T(cu) = cT(u).
The transformation T: is a linear if and only if
T( cu + dv) = cT(u) + dT(v )
for all pairs of vectors u and v in V and all pairs of scalars c and d i.e. a transformation between two vector spaces is linear if, and only if, it preserves linear combinations of pairs of vectors.
General form:
Example:
Determine whether the following transformations are linear.
a.
b.
Example:38
Define a transformation T: by: .
Find the images under T of , , and
Example:Find a vector whose image under T(x)=Ax is b and determine whether x is unique.
Example:Find all x in R4 that are mapped onto the zero vector by the transformation x Ax given:
Example:
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Let , , define a linear transformation T: that maps x into .
Find a matrix A such that T(x) is Ax for each x.
HW- read 1.8 do p68 #1-19 oddLinear Algebra Name:____________________________________Lesson- Matrix of a Transformations (1.9)
Date:_____________________________________
Objective: To learn how to use matrices to perform transformations in a vector space.
Theorem: Let T: be a linear transformation. Then there exists a unique matrix A such that:
for all x in
Where A (known as the standard matrix for the linear transformation) is the m x n matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix in . Alternatively,
Example:
Find the standard matrix A for the dilation transformation T(x) = 4x, for x in .
Example:40
Let T: be the transformation that rotates each point in about the origin through an angle , with counterclockwise rotation for a positive angle. Find the standard matrix of this transformation.
Describing a linear transformation T : 2 2 in geometrical terms :
Transformation rules from Geometry:
Reflections:
Dilations:
Translations:
Rotations:
*note- in the unit circle and
Reflections:
Example: Find the image of the vector (2, 3) under the transformation T(x, y) = (x, -y). Draw the vectors on the same coordinate plane and describe what you observe.
Rotations:
Consider the effect of the transformation T on each standard unit basis vector of 2 given by
),sin ,(cos)( 1 eT and ).cos,sin())2
sin( ),2
(cos()( 2 eT
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Figure: y )( 2eT 2e )( 1eT x The matrix of the transformation is
cos sinsin cos
A ,
and)cossin ,sincos(),( yxyxyxT .
Example: Find the image of the vector (2, 2) under the transformation
)cossin ,sincos(),( yxyxyxT
with 2 . Draw the vectors on the same coordinate plane and describe what you observe.
Expansion(Compression) in the x-direction
Consider the effect of the transformation T on each standard unit basis vector of 2 given by ),0 ,()( 1 ceT and 0 ),1 ,0()( 2 ceT
The matrix of the transformation is
1 00 c
A ,
andT(x, y) = (cx, y).
If c > 1, the transformation is an expansion If 0 < c < 1, the transformation is a compression If c = 1, the transformation is the Identity transformation
Shear in the x-direction
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Consider the effect of the transformation T on each standard unit basis vector of 2 given by ),0 ,1()( 1 eT and ).1 ,()( 2 ceT
The matrix of the transformation is
1 0 1 c
A ,
andT(x, y) = (x + cy, y).
Example: Find the image of the vector (2, 3) under the transformation T(x, y) = (x + 2y, y). Draw the vectors on the same coordinate plane and describe what you observe.
Definition: The linear transformation is one-to-one if no two different vectors have the same image in W.
Definition: The transformation is onto if every vector in W is the image of at least one vector in V.
Examples: 1. The transformation defined by is one-to-one but is not onto. Why is
transformation not onto?
2. The transformation defined by is onto but is not one-to-one. Why is the transformation not one-to-one?
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Shortcut rules:
44
HW- read 1.9 do p78 #1-37 every other oddLinear Algebra Name:____________________________________Lesson- Business Applications (1.10)
Date:_____________________________________
Objective: To learn how to apply concepts of linear algebra to real-world scenarios.
Constructing a Nutritious Weight-Loss Diet
Example:
If possible, find some combination of nonfat milk, soy flour, and whey to provide the exact amounts of protein, carbohydrate, and fat supplied by the diet in one day.
Amounts (g) Supplied per 100 g of Ingredient Amounts (g) Supplied by Cambridge Diet in One DayNutrient Nonfat milk Soy flour Whey
Protein 36 51 13 33Carbohydrate 52 34 74 45Fat 0 7 1.1 3
Linear Equations and Electrical Networks
Kirchoff’s first law states that the total current coming into a node (or junction) equals the total current leaving a node. Kirchoff’s second law states that the net RI voltage drop around a loop must equal the sum of the component RI voltage drops within the loop. The second law is what we will focus on here:
Example:45
Determine the loop currents in the diagram below by constructing an appropriate matrix and solve using GC.
Difference Equations
A linear difference equation is also called a linear recurrence relation, because it can be used to compute recursively each yk from the preceding y-values.
Example:
In a certain region, about 6% of a city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2010, there were 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where xo is the initial population in 2010. Then estimate the populations in the city and in the suburbs two years later, in 2012.
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HW- read 1.10 do p86 #1-11 odd
LINEAR ALGEBRAChapter 2- Matrix Algebra
47
Linear Algebra Name:____________________________________Lesson- Matrix Operations (2.1)
Date:_____________________________________
Objective: To learn about different types of matrix operations and their properties.
Matrix Addition
Let and . A + B =
We define matrix addition by adding component-wise. Dimensions must be the same.
Example:
Also note that matrix subtraction is just addition of the scalar multiple of -1 applied to a matrix.
Shorthand Notation: We can represent an arbitrary entry of a matrix A by the designation
The ‘i’ stands for the ith row, the ‘j’ stands for the jth column. So, a12 means the 1st entry in the 2nd row, a22 means the 2nd entry in the 2nd row, and so on.
So, we write an arbitrary 2x2 matrix by: We write an aribrary 2x3 matrix by:
We write an arbitrary 3x3 matrix by: And we write an arbitrary mxn music (m rows, n columns) by:
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So, to revisit Matrix addition:
Scalar multiplication: Transpose of a matrix:The transpose of A is denoted as
Note: if A is a symmetric matrix, then .
Example:
Given: and
Find:
a. A + B b. 2A c. AT
Matrix multiplication:
Definition of dot product:The dot product or inner product of the n-vectors
and are .
Example:
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Let and . Then, .
Definition of matrix multiplication:
Example:
Let’s say we have the matrix A = and we want to multiply it by the column vector v = .
The result would be Av = as we learned in Unit 1.
Example:
.
AC=
Example:
AC=
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Example:
Let . Construct a 2 x 2 matrix B such that AB is the zero matrix. Use two different nonzero
columns for B.
Note:I. AB is not necessarily equal to BA.II. might be not equal to B.III. AB = 0, it is not necessary that A = 0 or B = 0.IV. , , . Also, is not necessarily equal to .V. Let A be a square matrix and is a identity matrix. Then, .
Examples:
I.
AB=
BA=
II.
AC=
BC=
III.
AB=
BA=
IV.
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V.
Theorem: (A + B)T = AT + BT (or, the transpose of a sum is the sum of the transposes).
Example:
Given: and
Find:(A + B)T
AT + BT
Other important propertiesLet A, B and C denote matrices whose sizes are appropriate for the following sums and products.A + B = B + Ax(A + B) = xA + xB, where x is any number(x+y)A = xA + yBAB does not always equal BAA(BC) = (AB)CA(BC) does not always equal (AC)B (for example, consider A = I)(AT)T = A
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(A + B)T = AT + BT
(xA)T = x AT
(AB)T = BT AT
Note: we define, for x a real number and A a matrix, xA to be the matrix whose entries are x times those of A.
HW- read 2.1 do p100 #1-11 odd, 23, 27Linear Algebra Name:____________________________________Lesson- Inverse Matrices (2.2, 2.3)
Date:_____________________________________
Objective: To learn how to find inverse matrices. To learn what makes a matrix invertible.
Example
Find if
GOAL: We will use Gaussian Elimination to get A to the identity matrix (ones on the main diagonal, zeros elsewhere). We will keep track of the Gaussian Elimination by acting on the Identity matrix.
Rule:
Essentially we will make the same moves on I that we make on A.
Example
Find if
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Shortcut Rule for Inverse of a 2x2 Matrix
Let . If , then A is invertible and .
If then A is not invertible.DeterminantFor a 2 x 2 matrix a determinant can be found by
Examples:
(1)6748
(2)2
21
121
(3)
2x3x3x8x
(4) Matrix . The determinant of A is -2 and . Find e, f, g, and h.
TheoremIf A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the unique solution x = b
Examples:(1) Megan paid $39 for 3 pounds of pistachios and 2 pounds of cashews. Tanya paid $23 for 2 pounds of
pistachios and 1 pound of cashews. Write a system of equations, where x represents the cost of a pound of pistachios and y represents the cost of a pound of cashews. Then write a matrix equation and solve.
(2) Frank and Juanita sold tickets for the charity fund-raiser. They sold both single tickets and 5-ticket books. Write the appropriate matrix equation and find the price of a single ticket and a book of tickets.
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Other important properties
AA-1 = I, the Identity matrix(A-1)T = (AT)-1
(AB)-1 = B-1 A-1
Elementary Matricesmatrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of invertible matrices.
TheoremAn n x n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A-1.
Algorithm for finding A-1
Row reduce the augmented matrix [A I]. If is row equivalent to I, then [A I] is row equivalent to [I A -1]. Otherwise, A does not have an inverse.
Example:
Find the inverse of the matrix, if it exists.
Properties of the Inverse:(1) The identity matrix I is invertible and I-1 = I.(2) If A is an invertible matrix then so also is A-1, and (A-1)-1 = A.(3) If A, B are invertible matrices, then so also is AB, and (AB)-1 = B-1A-1.(4) If A is an invertible matrix, then so also is Ak, and (Ak)-1 = (A-1)k.
(5) If A is an invertible matrix and if c is any non-zero constant, then cAc
A 1 11.
(6) If A is an invertible matrix, then so also is AT, and (AT)-1 = (A-1)T.(7) If A is an invertible matrix, and if AB = AC or if BA = CA, then B = C.(8) If A is an invertible matrix, then A-1 is unique.(9) If A is an invertible nn matrix, then Rank (A) = n.(10) If A is an invertible nn matrix, then A can be reduced to an nn identity matrix by means
of the elementary row operations.(11) If A is an invertible matrix, then A is a product of elementary matrices. An elementary
matrix is a matrix obtained from an identity matrix by means of exactly one elementary row operation.
(12) If A is an invertible matrix, then the matrix equation 55
(a) AX = 0 means that X = 0 is the unique solution (b) AX = B means that X = A-1B is the unique solution.
Invertible Matrix Characterization:Let A be an matrix. The following statements are equivalent:
A is invertible A is the row equivalent to the n x n identity matrix Every column of A is a pivot column The equation Ax = 0 has only the trivia solution The column vectors of A are linearly independent The linear transformations x Ax is one-to-one The equation Ax = b has at least one solution for each b in Rn
The columns of A span Rn
The linear transformation x Ax maps Rn onto Rn
There is an n x n matrix C such that CA = I There is an n x n matrix D such that AD = I AT is an invertible matrix
The above leads to the following property:
Let A, B are square matrices. If AB = I, then A and B are both invertible, with B = A-1 and A = B-1
Example
Use the Invertible Matrix Theorem to decide if A is invertible:
Example
Use the Invertible Matrix Theorem to decide if A is invertible:
HW- read 2.2, 2.3 do p109 #1-7 odd, 29-35 odd, p115 #1-9 oddLinear Algebra Name:____________________________________
56
Lesson- Matrix Factorizations (2.5)Date:_____________________________________
Objective: To learn how to identify factors of matrices.
Suppose that A can be factored into the product of a lower triangular Matrix L and an upper triangular matrix: A LU . Then, to solve the system of equations Ax b , it is enough to solve this problem in two stages:
Lz b solve for zUx z solve for x
We begin with a n n matrix A and search for matrices
such that A = LU.
L and U are not uniquely determined. For each i, we can assign a nonzero value to either iil of iiu (but not both). For example, one simple choice is to set 1iil for 1,2, ,i n , making the L matrix-factor lower triangular. We can also make U upper triangular ( 1iiu for each i).
In the case of a 4 x 4 matrix, for example, we would be looking at something like this:
Where * represents some non-zero value.
Algorithm for an LU Factorization1. Reduce A to an echelon form U by a sequence of row replacement operations, if possible.2. Place entries in L such that the same sequence of row operations reduces L to I.
Example: Find the LU factorization of A.
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60 30 2030 20 1520 15 12
A
Example:
Show that does not have an LU factorization.
Example:
Solve the equation Ax = b by using the LU factorization given for A.
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Example:
Find an LU factorization of each:
a. b.
Example
a. Compute the transfer matrix of the ladder network below
b. Design a ladder network whose transfer matrix is
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HW- read 2.5 do p129 #1-15 odd, 29Linear Algebra Name:____________________________________Lesson- Leontief I-O Model (2.6)
Date:_____________________________________
Objective: To learn about the Leontief Input-Output Model
The Leontief Input-Output Model
A modern economy is comprised of a number of different industries that interact with one another. For example, the steel industry provides material for making mining equipment, which is used in mining coal, which is then used in the production of steel. This kind of interaction among various economic components makes a planned economy very difficult to manage.
To analyze an interacting economy, the Russian-Born American economist, Wassily Leontief, developed a mathematical modeling procedure called input-output analysis. Leontief’s model has proved extremely useful to economists in a variety of ways, such as in forecasting production needs and in predicting the effects of price changes. In recognition of his pioneering work on input-output analysis, Leontief was awarded the Nobel Prize for Economics in 1973.
To describe the basic ingredients of Leontief’s model, we assume the economic system we wish to study has n different sectors S1, S2,…, Sn. Typical sectors in this context include agriculture, manufacturing, transportation, energy, law enforcement, and so on. Portions of the output of each sector are used as input by the others. A sector may also use some of its own output as input for further production. For instance, some farmers use part of the alfalfa they grow to feed their livestock.
To represent these interactions in matrix form, we let number of units from sector Si consumed in the production of one unit of output by sector Sj and then form the input-output matrix (also called a technology matrix)
For example, consider a simple economy with three sectors: computers (C), services (S), and energy (E). Suppose this economy has the following input-output matrix.
The entries in each column represent the number of input units that are needed to produce one unit of output. For instance, the numbers in the S (services) column tell us that to produce 1 unit of services, we need 0.3 units
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form the computer sector, 0.5 units from the energy sector, and 0.1 units from the services sector itself. If the units are expressed in monetary terms, this means that, for each $1 of services that are produces, the services sector uses 30 cents’ worth of computer products, 10 cents’ worth of services, and 50 cents’ worth of energy.
The Leontief Input-Output Model (aka Production Equation)
Theorem
Let C be the consumption matrix for an economy, and let d be the final demand. If C and d have nonnegative entries and if each column sum of C is less than 1, then (I – C)-1 exists and the production vector x = (I – C)-1 d has nonnegative entries and is the unique solution of x = Cx + d.
The fundamental problem of input-output analysis is to determine the output levels for each sector that will result in a desired level of external demand. Here is a typical problem.
Example: Determining Production Needed to Satisfy Known Demand.
We have just considered a simple economy in which there are three sectors, computers (C), services (S), and energy (E), with the input-output matrix
Suppose in this economy that there is a demand for 70 units of computers, 100 units of services, and 150 units of energy. What level of production in each sector will satisfy this demand?
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HW- read 2.6 do p136 #1-9 odd
LINEAR ALGEBRAChapter 3- Determinants
62
Linear Algebra Name:____________________________________Lesson- Intro to Determinants (3.1)
Date:_____________________________________
Objective: To learn the definition of a determinant.
Definition of permutation: Let be the set of integers from 1 to n. A rearrangement of the elements of is called a permutation of .
Example:
Let . Then, 123, 231, 312, 132, 213, 321 are 6 permutations of .
Note: there are n! permutations of .
no inversion. 1 inversion (32) 1 inversion (21) 2 inversion (21, 31) 2 inversion (31, 32) 3 inversion (21, 32, 31)
Definition of even and odd permutations:When a total number of inversions of is even, then is called an even permutation. When a total number of inversions of is odd, then is called an odd permutation.
Determinants:
Let be an square matrix. We define the determinant of A (written as or ) by:
where is a permutation of .
If is an even permutation, then . If is an odd permutation, then .
Note: . Any two of are not in the same row and also not in the same column.
Proof of determinant:
63
There are two terms in the determinant of A:
Example:
.
Proof of determinant:
.
Then, there are 6 terms in the determinant of A,
Example:
Definition of determinant
For , the determinant of an matrix [ ] is the sum on n terms of the form , with plus and minus signs alternating, where the entries are from the first row of A. In symbolic form:
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We define, as the matrix that results if you were to cross off row 1 and column j. This allows us to define the determinant of an in terms of an . As long as we account for all possible
matrices, we will have a set of matrices that linearly combine to give the true determinant.
Example:
Using the formula above compute the determinant of:
Definition
Let [ ], the (i, j)-cofactor is given by:
Theorem
The determinant of an matrix A can be computed by a cofactor expansion across any row or down any column.
The expansion across the ith row using cofactor is:
The cofactor expansion down the jth column is:
Example:
Use a cofactor expansion across the third row to compute det A, where
Example:
Compute det A, where
65
Theorem
If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.
HW- read 3.1 do p167 #1-37 every other oddLinear Algebra Name:____________________________________Lesson- Properties of Determinants (3.2)
Date:_____________________________________
Objective: To learn various properties and characteristics associated with determinants.
Properties of Determinants:Let A be a matrix.
(a) 66
(b) If two rows (or columns) of A are equal, then .(c) If a row (or column) of A consists entirely of 0, then
Example:
Let .
Find each:(a) and
(b)
(c)
Properties of Determinants continued:(d) If B result from the matrix A by interchanging two rows (or columns) of A, then .(e) If B results from A by multiplying a row (or column) of A by a real number c,
for some i, then .(f) If B results from A by adding to ,
i.e., (or ), then
Example:
(d) Let
where B results from A by interchanging the first two rows of A. Find and
Example:
(e) Let where . Find and
Example:
(f) Let
where . Find and
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Properties of Determinants continued:
(g) If a matrix [ ] is upper triangular (or lower triangular), then .(h)
If A is nonsingular, then .
(i)
Example:
(g) Let . Find
Example:
(h) Let . Find and
Example:
(i) Let Find and
Example:
Let if , compute:
a. det [ ] b.
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HW- read 3.2 do p175 #1-29 every other oddLinear Algebra Name:____________________________________Lesson- Cramer’s Rule (3.3)
Date:_____________________________________
Objective: To learn what Cramer’s rule is and how to apply it.
Cramer’s Rule
Let A be an invertible matrix. For any b in Rn, the unique solution x of Ax = b has entries given by
Solutions of SystemsIf D ≠ 0, the system is consistent and has one unique solution.
If D = 0 and at least one numerator determinant is 0, the system is dependent and has infinitely many solutions.
If D = 0 and neither numerator determinant is 0, the system is inconsistent and has no solution.
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Examples
Use Cramer’s rule to solve:
(1) (2)
(3) (4)
(5) (6)
Theorem:
Let A be an invertible matrix. Then
Where (aka adjugate or classical adjoint) represents the cofactor expansion of A.
This is arrived at by establishing the jth column of as a vector x that satisfies Ax = ej where ej is the jth column of the identity matrix and the ith entry of x is the (i, j)-entry of
Example:
Find the inverse of the matrix
70
Theorem:
If A is a matrix, the area of the parallelogram determined by the columns of A is . If A is a matrix, the volume of the parallelepiped determined by the columns of A is .
Example:
Determine the area of the parallelogram whose vertices are at (-2, -2), (0, 3), (4, -1), and (6, 4).
Example:
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 4, 0), (-2, -5, 2), and (-1, 2, -1).
Theorem:
Let be the linear transformation determined by a matrix A. If S is a parallelogram in , then
If T is determined by a matrix A, and S is a parallelepiped in , then
Example
71
Let S be the parallelogram determined by the vectors . Compute the area
of the image under the mapping x Ax.
HW- read 3.3 do p185 #1-27 odd
72
LINEAR ALGEBRAChapter 4- Vector Spaces
Linear Algebra Name:____________________________________Lesson- Vector Spaces and Subspaces (4.1)
Date:_____________________________________
Objective: To learn the different properties and characteristics of vector spaces and subspaces.
Properties of Vectors in
Definition of vector:
Recall: An n-vector in is an matrix.
Definition of Vector Space
Definition: A real vector space V is a set of elements together with two operations, addition and scalar multiplication, satisfying the following properties:
Let u, v, and w be vectors in , and let c and d be scalars. Then, 1. is in V2.3.4.
73
5.6. is in V7.8.9.10.
Properties
Let u be any element of a real vector space V. Then,(a) (b) (c) (d)
Subspace
Definition of subspace:W is called a subspace of a real vector space V if 1. W is a subset of the vector space V.2. W is a vector space with respect to the operations in V.
Properties/Conditions
W is a subspace of a real vector space V 1. The zero vector of V is in W. If this property is not verified, then properties 2 and 3 are irrelevant.2. If u and v are any vectors in W, then .3. If c is any real number and u is any vector in W, then .Example:
the subset of consisting of all vectors of the form, , together with standard addition and
scalar multiplication. Is a subspace of ?
Solution:If
,
Then,
(1): and/or can = 0 therefore the zero vector of V is in W
(2): .
(3): .
is a subspace of .
Example:
74
Let the real vector space V be the set consisting of all matrices together with standard addition and scalar multiplication. Let the subset of V consisting of all diagonal matrices. Is a subspace of V?Solution:Let
, , and .
(1): and/or can = 0 therefore the zero matrix of V is in W
(2): since is still a diagonal matrix.
(3): since is still a diagonal matrix.
is a subspace of V.Example:
the subset of consisting of all vectors of the form,
,
together with standard addition and scalar multiplication. Is a subspace of ?
Theorem:
If are in a vector space V, then is a subspace of V.
Example:
Given and in a vector space V, let H = . Show H is a subspace of V.
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Example:
For what values of h will y be in the subspace of spanned by 1v , , if
, ,
HW- read 4.1 do p195 #1-19 oddLinear Algebra Name:____________________________________Lesson- Null, Column Spaces and LTs (4.2)
Date:_____________________________________
Objective: To become familiar with the concepts and properties associated with subspaces.
Null Space
The null space of an matrix A, written as Nul A, is the set of all solutions of the homogeneous equation Ax = 0. In set notation, Nul A = {x: x is in and Ax = 0}
Example:
Let . Determine if u belongs to the null space of A.
76
Theorem:
The null space of an matrix A is a subspace of . Also, the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of .
Column Space
The column space of an matrix A, written as Col A, is the set of all linear combinations of the columns of A. If , then
Theorem
The column space of an matrix A is a subspace of
The column space of an matrix A is all of iff the equation Ax = b has a solution for each b in .
Example:
.
a. If the column space of A is a subspace of , what is k?b. If the null space of A is a subspace of , what is k?c. Find a nonzero vector in Col A and a nonzero vector in Nul A.
d. Let
i. Determine if u is in Nul A. Could u be in Col A?ii. Determine if v is in Nul A. Could v be in Col A?
Kernel and Range of a Linear Transformation
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Definition
A linear transformation T from a vector space V into vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W such that
(a) T( u + v) = T(u) + T(v) for all u,v in V, and (b) T(cu) = cT(u) for all u in V and all scalars c.
The kernel (or null space) of T is the set of all u in V such that T(u) = 0. The range of T is the set of all vectors in W of the form T(x) for some x in V.
Example:
Define T: C[0,1]C[0,1] as follows: For f in C[0,1], let T(f) be the antiderivative F of f such that F(0)=0. Show that T is a linear transformation, and describe the kernel of T.
78
HW- read 4.2 do p206 #1-33 oddLinear Algebra Name:____________________________________Lesson- Linearly Independent Sets; Bases (4.3)
Date:_____________________________________
Objective: To learn about the relationships inherent in linearly independent sets. To learn what the term “basis” represents and how it is applied.
Definition of “basis”:The vectors in a vector space V are said to form a basis of V if (a) span V (i.e., ).(b) are linearly independent.
Example:
. Are and a basis in ?
Solution:
and form a basis in since (a) (b) and are linearly independent
Example:
. Are and a basis in ?
79
Solution:
and are not a basis of since and are linearly dependent: .Note that .
Example:
Are and a basis in ?
Solution:
and are not a basis in since and are linearly independent,
.
Example:
Let . Are S a basis in ?
Important result:If is a basis for a vector space V, then every vector in V can be written in an unique way as a linear combination of the vectors in S.
Example:
If corresponds to the standard basis where and
determine the basis decomposition of .
Example:
80
321321 ,, and ,100
,010
,001
eeeSeee
. S is a basis of .
Then, for any vector , is uniquely determined.
Important result:Let be a set of nonzero vectors in a vector space V and let . Then, some subset of S is a basis of W. How to find a basis (subset of S) of W:
Step 1: Form equation .
Step 2: Construct the augmented matrix associated with the equation in step 1 and transform this augmented matrix to the reduced row echelon form.
Step 3: The vectors corresponding to the columns containing the leading 1’s form a basis. For example, if k = 6 and the reduced row echelon matrix is
,
then the 1st, the 3nd, and the 4th columns contain a leading 1 are a basis of .
The pivot columns of a matrix A form a basis for Col A.
Example:
Let and . Find the subsets of S which form a basis of
.
81
HW- read 4.3 do p213 #1-19 oddLinear Algebra Name:____________________________________Lesson- Coordinate Systems (4.4)
Date:_____________________________________
Objective: Using bases to create a coordinate system for V.
The Uniqueness Representation Theorem:
Let be a basis for vector space V. Then for each x in V, there exists a unique set of scalars such that
If is a basis for V and x is in V, then the coordinates of x relative to the basis (or the -coordinates of x) are the weights such that
Therefore:
Example:
Find the vector x determined by the given coordinate vector and the given basis .
Example:
Find the coordinate vector of x relative to the given basis
a.
82
b.
Example:
Use coordinate vectors to verify that the polynomials are linearly dependent in P2.
Example:
Use an inverse matrix to find for the given x and .
83
HW- read 4.4 do p222 #1-13 oddLinear Algebra Name:____________________________________Lesson- Dimensions (4.5)
Date:_____________________________________
Objective: to learn the definition of dimension as it applies to vector spaces.
Definition of dimension:
The dimension of a vector space V is the number of vectors in a basis for V.
Example:
is basis for .
The dimension of R3 is 3.
Important result:Let V be an n-dimensional vector space, and let be a set of n vectors in V.(a) If S is linearly independent, then S is a basis for V.(b) If S spans V, then S is a basis for V.
Example:
Is a basis for ?
Solution:Since is a 3-dimensional vector space, not like in the previous example, we only need to examine whether S is linearly independent or S spans . We don’t need to examine S being both linearly independent and span V.
84
Example:
Is a basis for ?
Solution:
Since is a 3-dimensional vector space, we only need to examine whether S is linearly independent or if S spans . Because
,
are linearly independent. Therefore, are a basis of
Dimensions of Nul A and Col A
The dimension of Nul A is the number of free variables in the equation Ax = 0, and the dimensions of Col A is the number of pivot columns in A.
Example:
Find the dimensions of the null space and the column space of A.
85
HW- read 4.5 do p222 #1-17 oddLinear Algebra Name:____________________________________Lesson- Rank (4.6)
Date:_____________________________________
Objective: To learn about the rank of a matrix
Rank of a Matrix:
Recall:Let
.
The ith row of A is ,
and the jth column of A is
Definition of row space and column space:
,which is a vector space under standard matrix addition and scalar multiplication, is referred to as the row space.
Similarly, ,which is also a vector space under standard matrix addition and scalar multiplication, is referred to as the column space.
Definition of row equivalence:A matrix B is row equivalent to a matrix A if B result from A via elementary row operations.
Example:
Through various row operations we determine that are all row equivalent to AImportant Result:If A and B are two row equivalent matrices, then the row spaces of A and B are equal.
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How to find the bases of the row and column spaces:
Suppose A is a matrix. Then, the bases of the row and column spaces can be found via the following steps.
Step 1: Transform the matrix A to the matrix in reduced row echelon form.
Step 2: The nonzero rows of the matrix in reduced row echelon form give the basis of the row space of A. The columns containing the leading 1’s in rref give the basis of the column space of A.
For example, if n = 6 and the reduced row echelon matrix is
,
then the 1st, the 2nd, and the 3rd rows contain nonzero entries and the 1st, the 3rd, and the 4th columns contain a leading 1 so that means form a basis of the row space of A and
form a basis of the column space of A.
Note:To find the basis of the column space is to find to basis for the vector space
.
Example:
Let .
Find the bases of the row and column spaces of A.
Definition of row rank and column rank:
The dimension of the row space of A is called the row rank of A and the dimension of the column space of A is called the column rank of A.
Example:
87
Since the basis of the row space of A in the example above is ,
the dimension of the row space is 3 and therefore the row rank of A is 3.
Similarly,
is the basis of the column space of A making the dimension of the column space is 3 and therefore the column rank of A is 3.
Important Result:The row rank and column rank of the matrix A are equal.
Definition of the rank of a matrix:Since the row rank and the column rank of a matrix A are equal, we only refer to the rank of A and write
.
Important Result:If A is a matrix , then:
Example:
Determine the bases, rank and nullity given and .
Important Result:Let A be an matrix.
A is nonsingular (invertible) if and only if .
A is nonsingular Ax = b has a unique solution
A is nonsingular Ax = 0 has a nontrivial solution88
Let A be an matrix. Then, Ax = b has a solution rank(A) = rank[A|b]
HW- read 4.6 do p236 #1-15 oddLinear Algebra Name:____________________________________Lesson- Changing Basis (4.7)
Date:_____________________________________
Objective: To learn how to change the basis in a vector space.
Change of Basis:
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Let and be bases for n-dimensional vector space.
Then there is a unique matrix such that
The columns of are the C-coordinate vectors of the vectors in basis B meaning:
A general procedure to find in Rn
Let and be bases in Rn, then can be found by the following two steps:
Step 1. Form the augmented matrix
.
Step 2: Transform this augmented matrix into the reduced row echelon matrix
The matrix in the right side of the reduced row echelon matrix is
Important result:Let and be bases for n-dimensional vector space V. Let be the
transition matrix from C-basis to B-basis and be the matrix from B-basis to C-basis.
Then, is nonsingular and
Example:
Let .
Find and
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Example:
Let B={b1,b2} and C={c1,c2} be bases for R2. Find and
HW- read 4.7 do p242 #1-9 oddLinear Algebra Name:____________________________________Lesson- Difference Equations (4.8)
Date:_____________________________________
Objective:
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Linear Algebra Name:____________________________________Lesson- Markov Chains (4.9)
Date:_____________________________________
Objective:
Markov chain:
Example:
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Suppose there are 2 competing products, E and N, in the market, for example, Explorer and Netscape. After one year, some customers will keep using the same product but some will switch to using the other product. The proportion of the customers who keep using the same product or switch to use the other product after one year is summarized in the following table:
E NKeep 1/4 1/3
Switch 3/4 2/3
Suppose of the customers use product E and the other of customers use product N in the beginning. Then,
1. What is the distribution after one year?2. What is the distribution after two years?3. What is the distribution as the market is said to be stable (i.e., the distribution of the market would be
constant forever).
[solution:]
Let the matrix E N
,
The first row contains the proportions of which the customers keep using product E and the ones switch to using product E. The second row contains the proportions of which the customers switch to using product N and the ones keep using product N. Let
be the vector of distribution in the beginning.
1.
After one year, of customers use product E while of customers use product N.
2.
After two year, of customers use
product E while of customers use product N.
3. Suppose is the distribution as the market is stable. Thus,
93
,
where is a identity matrix. The solutions for the homogeneous system are
.
Since .
General case:
Suppose there are n competing products, C1, C2,…,Cn in the market. A general Markov chain model can be employed. The proportion of the customers who keep using the same product or switch to use the other product after one year is summarized in the following table:
C1 C2 … CnC1 (after one
year)…
C2 (after one year)
…
Cn (after one year)
…
Also,
Let C1 C2 … Cn
,
be the matrix representing the proportions of which the customers keep using the original product and those switch to using the other products. Suppose, in the beginning, the proportions of the customers for using C1, C2,…, Cn are
and the vector of distribution is
.
Then, The vector of distribution after k years is
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k factors Suppose
is the vector of distribution as the market is stable. Then, can be found by solving the following homogeneous linear system,
.
95
Linear Algebra Name:____________________________________Lesson- Systems of Inequalities
Date:_____________________________________
Objectives: solve systems of inequalities graphically determine the coordinates of the vertices of a region formed by the graph of a system of
inequalities
Do Now: Solve the system of inequalities by graphing:
y > x2 – 3 2x + 3y < 3
96
(1) Examine: | y | ≤ 3 (2) Solve the system of inequalities:y ≤ -x + 1| x + 1 | < 3
(3) Find the coordinates of the vertices of the figure formed by:2x – y ≥ -1x + y ≤ 4x + 4y ≥ 4
(4) Solve the system of inequalities:y < -x2 – 1 x2 + y2 ≤ 9
97
x
y
x
y
x
y
x
y
(5) Find the coordinates of the vertices of the figure formed by:y ≤ xy ≥ -33y + 5x ≤ 16
(6) Solve the system of inequalities:x ≤ 1y < 2x + 1x + 2y ≥ -3
Linear Algebra Name:____________________________________Lesson- Max and Min of graphical regions
Date:_____________________________________
Objectives: find the maximum and minimum values of a function over a region
Do Now:(1) Graph the following system of inequalities:
x 5y -3x2y x + 7y x – 4
(2) Name the coordinates of the vertices of the feasible region.
98
x
y
x
y
x
y
(3) Find the maximum and minimum values of the given function for this region.
Introduction to Linear Programming:When solving problems with inequalities, the boundary lines are the for the situation or conditions given to the variables.
The intersection of the graphs of the system of inequalities is the .
When the graph of constraints creates a polygon, the region is (although, it is possible that a system of inequalities forms an region, where part of it is open).
The maximum or minimum value of a related function always occurs at one of the of the feasible region.
99
(1) A painter has exactly 32 units of yellow dye and 54 units of red dye. He plans to mix as many gallons as possible of two shades of orange, color A and color B. Each gallon of color A requires 4 units of yellow dye and 1 unit of red dye. Each gallon of color B requires 1 unit of yellow dye and 6 units of red dye.
(a) Write a system of inequalities to represent the number of gallons that can be mixed.
(b) Draw the graph showing the feasible region.
(c) List the coordinates of the vertices of the feasible region.
(d) Write a function for the total number of gallons.
(e) Determine the maximum number of gallons possible.
(2) The Cruiser Bicycle Company makes two styles of bicycles: the Traveler, which sells for $250, and the Tourister, which sells for $600. Each bicycle has the same frame and tires, but the assembly and painting time required for the Traveler is only 1 hour, while it is 3 hours for the Tourister. There are 300 frames and 360 hours of labor available for production. How many bicycles of each model should be produced to maximize revenue?
100
(3) Graph the following system of inequalities:
x – 3y -75x + y 13x + 6y -93x – 2y -7f(x, y) = x – y
(a) Name the coordinates of the vertices of the feasible region.
(b) Find the maximum and minimum values of the given function for this region.
101
x
y
Linear Algebra Name:____________________________________Lesson- Applications of Linear Programming
Date:_____________________________________
Objective: solve real-world applications with linear programming
(1) Denise is a glass blower who can form 8 simple vases or 2 elaborate vases in an hour. In a work shift of no more than 8 hours, she can form at most 40 vases.
(a) Let s represent the hours forming simple vases and e the hours forming elaborate vases. Write a system of inequalities involving the time spent on each type of vase.
(b) Draw the graph showing the feasible region.
(c) List the coordinates of the vertices of the feasible region.
(d) If Denise makes a profit of $30 per hour worked on simple vases and $35 per hour worked on elaborate vases, write a function for the total profit on the vases.
(e) Determine the number of hours Denise should spend on each type of vase to maximize profit. What is that profit?
102
e
s
(2) The designers at Frequency Fashions are crafting two new handbags, where the proceeds are going to charity. Both handbags will be lined with canvas and use leather for the handles. For the new Ingram handbag, the designers need 4 yards of canvas and 1 yard of leather. For the new Shaffer handbag, they need 3 yards of leather and 2 yards of canvas. The company has purchased 56 yards of leather and 104 yards of canvas.
(a) Write a system of inequalities to represent the number of handbags that can be produced.
(b) Draw the graph showing the feasible region.
(c) List the coordinates of the vertices of the feasible region.
(d) If Frequency Fashions plans to sell the Ingram handbag at a profit of $20 each and the Shaffer handbag at a profit of $35 each, write a function for the total profit on the bags.
(e) Determine the number of Ingram and Shaffer handbags that they need to make for a maximum profit. What is that profit?
103
y
x
(3) Logarithmic Landscapers is a company that has crews who mow lawns and prune shrubbery. This company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120.
(a) Write a system of inequalities to represent the above situation.
(b) Draw the graph showing the feasible region.
(c) List the coordinates of the vertices of the feasible region.
(d) Write a function for Logarithmic Landscapers’ total daily income.
(e) Find a combination of mowing lawns and pruning shrubs that will maximize the income that Logarithmic Landscapers receives per day for one of its crews. What is that income?
104
y
x
(4) A college arena in Exponential City sells tickets to students and to the public. Student tickets are $8 each and general public tickets are $32 each. The college reserves at least 5000 tickets for students. The arena seats 18,000.
(a) Write a system of inequalities to represent the above situation.
(b) Draw the graph showing the feasible region.
(c) List the coordinates of the vertices of the feasible region.
(d) Write a function to calculate the revenue from the student and general public tickets.
(e) Determine the number of each type of ticket that the college should sell to maximize revenue. What is that revenue?
105
y
x