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4.1 Introduction to Linear Spaces(a.k.a. Vector Spaces)
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Recall: A SubspaceA subspace of a linear space V is called a
subspace if:a) W contains the neutral element 0 of Vb) W is closed under additionc) W is closed under scalar multiplication
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Recall: What are all of the possible vector subspaces in R2?
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What are all of the possible vector subspaces in R2?
A. The zero VectorB. Any line passing through the originC. All of R2
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Linear Spaces aka Vector SpacesA linear Space is a set with two well defined
operations, addition and scalar multiplication. Here are the properties that must be satisfied
1. (f+g)+h = f+(g+h) Associative Property2. f+g=g+f Commutative Property3. There exists a neutral element such that f+n =fThis n is unique and denoted by 04. For each f in V there exists g such that f+g=05. k(f+g) =kf+kg Distributive Property6.(c+k)f = cf + kf, Distributive Property7.c(kf) = (ck)f 8. 1f = f
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RecallSubspace
• A subset W in Rn is a subspace if it has the following 3 properties
• W contains the zero Vector in Rn
• W is closed under addition (of two vectors are in W then their sum is in W)
• W is closed under scalar multiplication
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Example 11
Show that the differentiable functions form a a subspace
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Example 11 Solution
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What are all of the vector subspaces of R3?
A) The zero vectorB) Any line passing through the originC) Any plane containing the origin.D) All of R3
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Example 12
a) Is the set of all polynomials a subspace?
b) Is the set of all polynomials of degree n a subspace?
c) Is the set of all polynomials with degree < n a subspace?
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Solution to 12
a) yesb) No, not closed under addition Example:x2 + 3 and –x2 + xc) yes
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Consider the elements f1,f2,f3,…fn in a linear space V
1. We say that f1,f2,f3,…fn span V if every f in V can be expressed as a linear combination of f1,f2,f3,…fn
2. We say that f1,f2,f3,…fn are linearly independent if the equation c1f1+c2f2+c3f3+…cnfn =0 has only the trivial solution where c1= … = cn = 0
3. We say that f1,f2,f3,…fn are a basis for V if they are both linearly independent and span V that means that every f in V can be written as a linear combination of f=c1f1+c2f2+c3f3+…cnfn
The coefficients c1,c2, …cn are called coordinates of f with respect to the basis β =(f1,f2,f3,…fn )
The vector is called the coordinate vector of f denoted by [f]β
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Dimension
If a linear Space has a basis with n elements then , all of the other basis consist of n elements as well. We say that n is the dimension of V or
dim(V) =n
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Example 15
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Example 15 Solution
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Coordinates
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Finding a basis of a linear Space
1) A write down a typical element in terms of some arbitrary constants
2) Using the arbitrary constants as coefficients, express your typical element as a linear combination of some elements of V.
3) Verify that all the elements of V in this linear combination are linearly independent.
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Example 16
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Example 16 solution
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Example
Find a basis and the dimension for all polynomials of degree n or less
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Example Solution
A basis would be
1, x, x , x , …xThe dimension is n+1
2 3 n
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Find a basis for the set of all polynomials What dimension is the linear space containing the set of all
polynomials?
Note the answer is on the next slide
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A linear Space V is called Finite dimensional if has a (finite) basis f1,f2,f3,…fn so that we can define its dimension dim(V) = n Otherwise, the space is called infinite dimensional
Finite vs. Infinite Dimensionality
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Homework p 163 1-16 all 17-41 odd
• Q: What is the physicist's definition of a vector space?
• A: A set V such that for any x in V, x has a little arrow drawn over it.