CHAPTER 4LINEAR TRANSFORMATION AND MATRICES

Theorem 4.6If L: VW is a linear transformation, then range L is a subspace of W.Proof: Note that range L is not empty space L(Ov) = Ow.Let Y1 and Y2 be in the range L. Then Y1 = L(x1) and Y2 = L(x2) for some x1 and x2 in V.

Now Y1 + Y2 = L(x1) + L(x2) = L(x1 + x2) Y1 + Y2 is in the range L.also if c is scalar then cY1 = cL(x1)= L(cX1), so cY is in range L.

Definition 4.5If L:VW is a linear transformation, then the dimension of range of L is called the rank of L and the dimension of the kernel is called the nullity of L.

Example: Let L: R3 R3 defined by

L = . Is L onto? Find a basis for range

L.

Theorem 4.7 If L: V W is a linear transformation then dim(Ker L) + dim(range L) = dim V Proof: Let n= dim V and k = dim (Ker L). If k = nker L = V L(X) = 0, X V range L = {0} dim range L = 0

Suppose 1 k < n. Show: dim(range L) = n-k.

Let { X1, X2. , Xk} be a basis for Ker L, we can extend this to be a basis for V Say S ={ X1, X2. , Xk Xk+1, , Xn}

Prove: T = {L(Xk+1), L(Xk+2) , ,L(Xn)} is a basis for range L

i) Show: T spans range LLet Y be any vector in range L. Then Y = L(X) for some X in V. Since S is a basis for V thenX = c1X1 + c2X2 + + cnXn, where c1, c2 and cn are real numbers ThenY = L(X) = L(c1X1 + c2X2 + + cnXn) = c1L(X1) + c2L(X2) + + ckL(Xk) + ck+1L(Xk+1)+ + cnL(Xn) = ck+1L(Xk+1)+ + cnL(Xn) since X1, X2,..., Xk are in Ker L. T spans range L

ii) Show that T is linearly independent Suppose that ck+1L(Xk+1)+ ck+2L(Xk+2)+ +cnL(Xn)= 0w L(ck+1(Xk+1)+ ck+2(Xk+2)+ +cn(Xn))=0

Hence the vector ck+1Xk+1 + + cnXn is in Ker L, and we can write it as a linear combination of the vectors in the basis for Ker L

ck+1Xk+1 + + cnXn = d1X1 + d2X2 + + dkXk

where d1,d2, dk are real numbers d1X1 + d2X2 ++dk Xk - ck+1Xk+1 - ck+2Xk+2 -cn Xn =0 Since S is linearly independent since is a basis for V, then d1 = d2 = = dk = ck+1 = = cn = 0.

Corollary: Let L:VW is a linear transformation and let dim V = dim W.a)If L is one to one, then it is onto.b)If L in onto, then it is one one.

Example: 1. Let: R2 R2 be a linear transformation defined by L .

a) Find a basis kernel L.b) Is L one to one?c) Is L onto?d) Find a basis for range L?

2. Let L: R4 R3 be a linear transformation defined by L(x,y,z,w)= (x+y, y-z, z-w).a)Is L onto?b) Find the dimension of kernel of A.c) Verify thm 4.7

3. Let L: P2P2 be a linear transformation defined by L(at2+ bt + c) = (a+c)t2=(b+c)t.a) Is t2-t-1 in ker L?b) Is t2 + t-1in ker L?c) Is 2t2 t in range L?d) Find a basis for ker L?e) Find a basis for rangeL

Matrix of Linear TransformationDefinition 4.6Let V be an n-dimensional vector space with basis S={X1, X2, ,Xn} . If X = a1X1+a2X2++anXn is any vector in V, then the vector

[X]s =

in Rn is called the coordinate vector of X with respect to the basis S. The component of [X]s are called the coordinates of X with respect to S.

Example: Let S= , be a basis for

R3. Find the coordinate vectors of the following vectors with respect to S of

X =

Theorem 4.8 Let L: VW be a linear transformation of an n-dimensional vector space V into an m-dimensional vector space W and let S = {X1, X2, , Xn} and T={ Y1, Y2, , Yn } be bases for V and W, receptively. Then the m x n matrix A, whose jth column is the coordinate vector [L(Xj)]T of L(Xj) with respect to T is associated with L and has the following property: If Y = L(X) for some X in V, then[Y]T = A[X]S,where [X]s and [Y]T are the coordinate vectors of X and Y with respect to the respective bases S and T. Moreover A is the only matrix with this property.

Example: Let L: R2 R3

a. Find the matrix A with respect to bases S and T where

b. Use the matrix obtained in a) to compute for .

Definition 4.7:The matrix A of Theorem 4.8 is called the matrix of L with the respect to the bases S and T. Equation (2) is called the representation of L with respect to S and T. We also say that equation (2) represents L with respect to S and T.

Assignment: L: R2 R3 be defined by

.

Let S and T be the natural basis for R2 and R3, respectively. Also, let and be bases for

R2 and R3, respectively.

Find the matrix representing L with respect to S and T S and T

c) Compute using the definition of

L and using the matrices obtained in a) and b).

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