Linear Algebra Question

Problem #1 – Find the matrix representation of linear transformation relative to two ordered bases. 1. Let 3 2:T P P® be a linear transformation defined by ( )( ) (2 1)T p x p x¢= + (T maps a polynomial in 3P

to its derivative evaluated at (2 1)x+ ), and let 3 2( , , ,1)B x x x= and 2( , ,1)B x x¢ = and be ordered bases for 3P and 2P respectively.

a. Find the matrix representation A of T relative to ,B B¢ . b. Use A to compute 3 2(4 5 4 7)T x x x- + -

2. Let 3 3:T P P® be defined by ( ( )) ( ) 4 ( ) ( )T p x p x p x p x¢¢ ¢= - + and let

3 2( , , ,1)B x x x= be an ordered basis for 3P .

a. Find the matrix representation of T relative to B . b. Use your answer from part a. to compute 3 2(2 2 1)T x x x- + - . c. Let 2 3( ,1 , , )B x x x x x¢ = + + be a second ordered basis for 3P . Find the matrix

representation of T relative to ,B B¢ .

Problem # 2 – Given a transformation T, prove that T is linear, one to one, and onto.

1. Consider the function 21:T P R® defined by [ ](( ( )) (0), (2)T p x p p= .

a. Find (2 )T x- b. Prove T is linear. c. Prove T is one to one. Chapter 6.7 d. Find [ ]( )1 5,1T- e. Prove T is onto. Chapter 6.7

2. Let 22:T P R® be a transformation defined by ( ) [ ]2 2 ,T ax bx c a c b+ + = + , and let 2(1, , )B x x= and

1 0 ,

0 1 B

æ öé ù é ù ¢ = ç ÷ê ú ê ú

ë û ë ûè ø be ordered bases for 2P and

2R respectively.

a. Find the matrix representation A of T relative to ,B B¢ . (Hint: Think of 1 as 20 0 1x x+ + b. Show that T is not one to one. c. Find [ ]( )1 5,6T- (This is the preimage of [ ]5,6 in 2P .) d. Show that T is linear.

Problem # 3 – Determine the kernel, image, rank and nullity of a linear transformation.

1. Let 2:T P R® be given by ( ) 3

2

( ) ( )T p t p t dt -

= ò . a. Prove that T is a linear transformation. b. Find the kernel of T. c. Find the nullity of T and the rank of T. d. Is T an isomorphism? Explain.

2. Consider the linear transformation : n mT R R® defined by ( )T A=v v, where 1 2 1 1 2 4 3 6 1 2 1 7

A é ù ê ú= ê ú ê ú- -ë û

.

The reduced row-echelon form of A is 1 2 0 3 0 0 1 4 0 0 0 0

-é ù ê ú ê ú ê úë û

.

a. Determine the value of m. b. Determine the value of n. c. Determine a basis for the image of T. d. Determine a basis for ker T. e. Verify the rank equation for T.

Problem # 4 – Proof involving linear transformations. 1. Prove that if :T V W® is linear, then the ker(T) is a subspace of V.

2. Prove that 3P is isomorphic to 2,2M by defining a mapping 3 2,2:T P M® and showing that T is linear, one-to-one, and onto.

3. Let L be a linear transformation from a vector space V into a vector space W where { }ker( )L = 0 . If 1 2 3, ,v v v are linearly independent vectors in V, prove that 1 2 3( ), ( ), ( )L L Lv v v are linearly independent vectors in W. (Indicate in your proof where you use the fact that { }ker( )L = 0 .)

4. Prove that any matrix transformation is linear. 5. Prove that if V is a finite-dimensional vector space with ordered basis ( )B = 1 2 nb ,b ,…,b and

: nT V R® is a mapping defined by ( ) [ ]BT =v v , then T is an isomorphism. 6. If :T V W® is a linear transformation, then T is one-to-one if and only if

ker { }T = 0 . 7. Prove that if :T V W® is a linear transformation, then T is one-to-one if and only if ker { }T = 0 .

8. Prove that if :T V V ¢® is a linear transformation and W is a subspace of V, then the image of W is a subspace of V ¢.

9. See section 6.7 #5.

Problem # 5 – Find a change of basis matrix and use it to change coordinates.

1. Let ( )2, ,1B x x= and ( )2 2 2,2 2 1, 2B x x x x x x¢= - - + - be ordered bases for 2P (the vector space of polynomials of degree less than or equal to 2). a. Find the change of basis matrix P such that [ ] [ ]B BP ¢=v v for every 2PÎv . b. If 22 3 1x x= + -v , find [ ]B¢v c. Express 22 3 1x x= + -v as a linear combination of the vectors in B¢. Problem # 6 – Find the eigenvalues and eigenspaces of a linear transformation.

1. Let 1 0 1 0 1 1 1 1 0

A é ù ê ú= ê ú ê úë û

. Find the eigenvalues of A and bases its eigenspaces.