maths linear algebra
TRANSCRIPT
-
8/10/2019 Maths linear algebra
1/19
Singapore University of
Technology & Design
MATH 10.004Systems of Linear Equations
Cohort 1
Meyer, Sections 1.1-1.2
-
8/10/2019 Maths linear algebra
2/19
Linear Equations Singularity and Non-Singularity Summary
LEARNING OBJECTIVES
After this cohort you will be able to ...
be able to draw and think about planes in multiple dimensions
based on a linear equation.
understand the geometry of systems of linear equations in terms
of planes.
categorize a system of equations by the existence and number ofits solutions.
2
-
8/10/2019 Maths linear algebra
3/19
Linear Equations Singularity and Non-Singularity Summary
ACTIVITY 1: LINEAR EQUATIONS
Consider the equation: 2x`4y3z5.
1. Find a solution of this equation.
2. Subtract 2 from thexvalue of your solution and add 1 to the y
value. Is it still a solution?
3. Why is there more than one solution to this equation?
4. Find all solutions to this equation.
3
-
8/10/2019 Maths linear algebra
4/19
Linear Equations Singularity and Non-Singularity Summary
ACTIVITY 1: LINEAR EQUATIONS
Consider the equation: 2x`4y3z5.
1. Find a solution of this equation.
px, y, zq p2, 1, 1qis a solution.
2. Subtract 2 from thexvalue of your solution and add 1 to the y
value. Is it still a solution? Why?
px, y, zq p0, 2, 1qis still a solution because2p2q `4p1q 3p0q 0.
3. Why is there more than one solution to this equation?
There are 3 unknownspx, y, zqand only one equation*.
4. Find all solutions to this equation.
If we letyP R andzP R, thenx 12
p54`3q.
Thus all solutions are described by:`1
2 p54`3q, , .4
-
8/10/2019 Maths linear algebra
5/19
Linear Equations Singularity and Non-Singularity Summary
SYSTEM OF LINEAR EQUATIONS
Alinear systemof mequations innunknowns is written as
a11x1`a12x2` a1nxnb1
a21x1`a22x2` a2nxnb2...
am1x1`am2x2` amnxnbm.
which is a mathematical way of expressingmlinear equalityconstraints that thenvariablesxi, iP t1, . . . , nu, need to satisfy.
5
-
8/10/2019 Maths linear algebra
6/19
Linear Equations Singularity and Non-Singularity Summary
GEOMETRIC INTERPRETATION
a11x1`a12x2` a1nxnb1
a21x1`a22x2` a2nxnb2...
am1x1`am2x2` amnxnbm.
mhyperplanes inndimensions, each of the form
ai1x1`ai2x2` ainxnbi.
In R2 this is a line. In R3 this is a plane. A system of linear
equations representsmsimultaneous expressions of this form.
6
Li E i Si l i d N Si l i S
-
8/10/2019 Maths linear algebra
7/19
Linear Equations Singularity and Non-Singularity Summary
SOLUTIONS IN R2
a11x1`a12x2b1 is a line
a21x1`a22x2b2 is a line
a11x1`a12x2b1
x2
a11
a12 x1`
b1
a12
x1
x2
7
Li E ti Si l it d N Si l it S
-
8/10/2019 Maths linear algebra
8/19
Linear Equations Singularity and Non-Singularity Summary
SOLUTIONS IN R2
The system can have either:
x1
x2
One Solution
x1
x2
No Solutions
x1
x2
Infinite Solutions
8
Linear Equations Singularity and Non Singularity Summary
-
8/10/2019 Maths linear algebra
9/19
Linear Equations Singularity and Non-Singularity Summary
ACTIVITY 2: GEOMETRIC INTERPRETATION (10 MINUTES)
Consider the linear system:
2xy1
x`y5
Draw the linear system in R2 using the geometric interpretation of
hyperplanes.
Solve forpx, yqand indicate how the solution appears in your figure.
9
Linear Equations Singularity and Non Singularity Summary
-
8/10/2019 Maths linear algebra
10/19
Linear Equations Singularity and Non-Singularity Summary
ACTIVITY 2: GEOMETRIC INTERPRETATION
x
y 2xy1
x`y5
px, yq p2, 3q
Theuniquesolution to
"2xy 1x`y 5
* is given by
px, yq p2, 3q, the point where the two lines cross.
10
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
11/19
Linear Equations Singularity and Non Singularity Summary
SINGULARITY AND NON-SINGULARITY
A system of linear equations can be either: Consistent
Unique solution (Non-Singular) Infinite number of solutions (Singular)
Inconsistent No solution (Singular)
A system of linear equations can be:
Underdetermined(underconstrained) more unknowns (variables) than equations (constraints)
Overdetermined(overconstrained)
more equations (constraints) than unknowns (variables)
11
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
12/19
Linear Equations Singularity and Non Singularity Summary
ACTIVITY 3: SOLUTION GEOMETRY (5 MINUTES)
The following plots show three planes in R3, where each plane
corresponds to one of the equations in a system of 3 linear equations
in 3 unknowns.
For each plot, specify whether the system described is consistentor
inconsistent, and if consistent specify the number of solutions.
12
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
13/19
q g y g y y
ACTIVITY 3: SOLUTION GEOMETRY
For each plot, specify whether the system described is consistentor
inconsistent, and if consistent specify the number of solutions.
Consistent? # Solutions? Singular?
1 Inconsistent None Singular
2 Consistent Infinite Singular
3 Inconsistent None Singular
4 Inconsistent None Singular
5 Consistent Unique Nonsingular
13
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
14/19
q g y g y y
ACTIVITY 4: PLANES IN R4 (15 MINUTES)
Consider the following three hyperplanes in R4:
u`v`w`z6
u`w`z4
u`w2
1. Describe the intersection of the hyperplanes.
2. Is the intersection a line, a point, or the empty set?
3. What is the intersection if a 4th plane,u 1, is included?
4. Find a 4th equation, which results in no solution.
14
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
15/19
ACTIVITY 4: PLANES IN R4
u`v`w`z6 (1)u`w`z4 (2)
u`w2 (3)
1. Describe the intersection of the hyperplanes.(2)-(3) yieldsz2.(1)-(2) yieldsv2.
Hence, our system simplifies to the equivalent systemvz2
andu`w2.
2. Is the intersection a line, a point, or an empty set?
The coordinates ofvandzare fixed anduandware related via
u`w2. This describes a line.
15
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
16/19
ACTIVITY 4: PLANES IN R4
u`v`w`z6u`w`z4
u`w2
3. What is the intersection if a 4th plane,u 1, is included?
With this additional constraint we have vz2 (as before) andu 1 andw3. Hence, the intersection now describes apoint.
4. Find a 4th equation, which results in no solution.
u`w5 would be such an equation since it contradictsu`w2.
16
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
17/19
ACTIVITY 5: SINGULARITY IN R3 (10 MINUTES)
Consider the following system in R3:
u` v` w2
u`2v`3w1
v`2w0
Explain why the system is inconsistent by finding a combination of thethree equations that adds to 0 1.
Replace the zero on the RHS to allow the equations to have solutions,
and what is one of the solutions?
17
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
18/19
ACTIVITY 5: SINGULARITY IN R3
Consider the following system in R3:
u` v` w2 (1)
u`2v`3w1 (2)
v`2w0 (3)
Explain why the system is inconsistent by finding a combination of thethree equations that adds to 0 1.
Replace the zero on the RHS to allow the equations to have solutions,
and what is one of the solutions?
Combining (1)-(2)+(3) yields 0 1. Thus the system is inconsistentand singular.
Consistency (hence a solution) is achieved by setting the zero on the
RHS to1. In this case, one solution is u3,v 1, andw0.
18
Linear Equations Singularity and Non-Singularity Summary
-
8/10/2019 Maths linear algebra
19/19
SUMMARY
The geometric interpretation of linear equations corresponds to
intersecting hyperplanes.
A linear system can have no solutions, a unique solution, or an
infinite number of solutions.
19