Answer: For Exercises 2730, suppose thatA =abcde f is the matrix of T : V W with respect

For Exercises 14, find v given the coordinate vector vG with respect to the basis G. vG = 4 1  G ; G = 3 2  , 1 4  2.

For Exercises 14, find v given the coordinate vector vG with respect to the basis G. vG = 2 5  G ; G = {3x + 1, 2x 4}

For Exercises 14, find v given the coordinate vector vG with respect to the basis G. vG = 1 0 3  G ; G = ' x2 x + 3, 3x2 + 4, 5x 2

For Exercises 14, find v given the coordinate vector vG with respect to the basis G. vG = 1 2 3 1 G ; G = 1 1 2 1 , 1 2 2 2 , 0 2 1 0 , 2 1 0 3 For

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 8 9  ; G = 2 0  , 0 3  6.

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 14x + 15; G = {2x, 5}

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 12x2 15x + 30; G {4x2, 3x , 5}

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 6 6 20 7  ; G = 2 0 0 0 , 0 3 0 0 , 0 0 5 0 , 0 0 0 1 9.

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 5 5  ; G = 1 2  , 3 1  10

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 11x; G = {3x + 2, 2x 5}

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 9x + 1; G = ' x2 1, 2x + 1, 3x2 1

For Exercises 512, find the coordinate vector of v with respect to the given basis G. v = 0 3 1 1  ; G = 1 0 1 0 , 0 2 1 1 , 1 1 0 1 , 0 0 1 1

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 1 3 2 1  ; vG = 4 3  G ; Q = 1 1  , 2 3 

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 4 0 3 1 ; vG = 2 2  G ; Q = {3x 2, x + 5}

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 112 013 011 ; vG = 1 3 4  G ; Q = ' x2 2x, x2 + x + 4, 3x + 1

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 1 31 4 01 1 1 0 ; vG =  2 0 1  G ; Q =  3 1 2  , 2 4 0  , 1 2 3

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 043 2 1 3 2 2 1  ; vG =  2 3 1  G ; Q = ' sin(x), cos(x), ex

For Exercises 1318, A is the matrix of linear transformation T : V W with respect to bases G and Q, respectively. Find T(v) for the given vG. A = 0011 1010 2001 1100 ; vG = 3 2 1 1 G ; Q = 1 0 1 2 , 2 1 0 2 , 1 4 2 1 , 3 1 2 4 For

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T a b  = bx a; G = {e1, e2}; Q = {x, 1}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T(ax +b) = ax2 +(a +b)x b; G = {x, 1}; Q = {x2, x, 1}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T( f (x)) = f  (x); G = {x2, x, 1}; Q = {x, 1}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T( f (x)) = x f  (x) + f (0); G = {x2 + 3, x 2, x2 + x}; Q = {x2, x, 1}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T a b  = b a a + 2b  ; G = {e1, e2}; Q = 1 1  , 1 2 

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T(ax + b) = (b a)x (a + b); G = {x, 1}; Q = {x + 1, x + 2}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T a b  = bx + a + b; G = 2 1 3 2  ; Q = {5x + 3, 2x + 1}

For Exercises 1926, find the matrix Aof the linear transformation T : V W with respect to bases G and Q, respectively. T a b c d = a + b b + c c + d  ; G = 1 0 0 0 , 1 1 0 0 , 1 1 1 0 , 1 1 1 1 ; Q = 1 0 0  , 1 1 0  , 1 1 1 !

For Exercises 2730, suppose that A = abc de f  is the matrix of T : V W with respect to bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T with respect to the given bases H and R. (a) H = {g1, g2, g3}, R = {2q1, q2} (b) H = {3g1, g2, g3}, R = {q1, q2}

For Exercises 2730, suppose that A = abc de f  is the matrix of T : V W with respect to bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T with respect to the given bases H and R. (a) H = {g3, g2, g1}, R = {q1, q2} (b) H = {g1, g2, g3}, R = {q2, q1}

For Exercises 2730, suppose that A = abc de f  is the matrix of T : V W with respect to bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T with respect to the given bases H and R. (a) H = {g3, g1, g2}, R = {q1, q2} (b) H = {g1, g3, g2}, R = {q2, q1}

For Exercises 2730, suppose that A = abc de f  is the matrix of T : V W with respect to bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T with respect to the given bases H and R. . (a) H = {g1, 5g2, g3}, R = {q1, 2q2} (b) H = {g2, g1, 3g3}, R = {4q2, q1}

Suppose that T : P1 P1 has matrix A = 1 3 0 1 with respect to the basis G = {x + 1, x 1} for the domain and Q = {1, x}for the codomain. Use the inverse of A to find T1(x).

Suppose that T : P1 P1 has matrix A = 2 7 1 3 with respect to the basis G = {2x + 1, 3x 1} for the domain and Q = {1, x + 3} for the codomain. Use the inverse of A to find T1(x + 2).

Suppose that T : R2 P1 has matrix A = 2 1 3 2 with respect to the basis G = 1 3  , 2 1  for the domain and Q = {x, 2x + 1} for the codomain. Use the inverse of A to find T1(x + 1). 34

Suppose that T : R2 P1 has matrix A = 1 3 1 4 with respect to the basis G = 2 5  , 1 3  for the domain and Q = {3x, 2x 1} for the codomain. Use the inverse of A to find T-1 (x-2).

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. An element v and basis G of P2 such that vG =  2 0 3  G

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. An element v and basis G of R22 such that vG = 1 7 4 4 G

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. A basis G of R2 such that 3 4  G = 7 5 

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. A basis G of P1 such that 3 5  G = 4x 11

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. Vector spaces V and W, bases for each, and a linear transformation T : V W that has matrix A = 212 031 with respect to the bases.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. Prove your claim. Vector spaces V and W, bases for each, and a linear transformation T : V W that has matrix A =  6 1 3 3 2 5 with respect to the bases.

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. The matrix of any linear transformation between finitedimensional vector spaces must be square.

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. The matrix of a linear transformation T : V W is unique.

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. If a b  G is the coordinate vector of a vector with respect to a basis G, then 2a 3b  G is the coordinate vector with respect to G of some vector. 4

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. If v = a b  G for G = {g1, g2}, then v = b a  G for G = {g2, g1}. 4

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. If 0 is the zero vector of a finite-dimensional vector space V, then 0V = 0 . . . 0 G for every basis G of V.

TRUE OR FALSE For Exercises 4146, determine if the statement is true or false, and justify your answer. . If G and H are two distinct bases for a finite-dimensional vector space V, then vG and vH cannot have the same entries for any element v of V.

Let G be a basis for a vector space V of dimension m. Show that a set of vectors {v1, ... , vk } is linearly independent in V if and only if the coordinate vectors {[v1]G, ... , [vk ]G} are linearly independent in Rm.

LetG be a basis for a vector spaceV, and suppose that vG = wG. Prove that v = w.

Let G be a basis for a vector space V of dimension m. Show that a linear combination of vectors v1, ... , vk is equal to v in V if and only if there is a linear combination of the coordinate vectors [v1]G, ... , [vk ]G that is equal to the coordinate vector vG in Rm.

Suppose that Ais the matrix of linear transformation T : V W with respect to bases G and Q, respectively. (a) Show that v is in the kernel of T if and only if vG is in the null space of A. (b) Show that w is in the range of T if and only if wQ is in the column space of A.

Suppose that Ais the matrix of linear transformation T : V V with respect to basis G for both the domain and codomain. Let T2(v) = T T(v) = T(T(v)) denote the composition of T with itself. (a) Show that A2 is the matrix of the linear transformation T2 : V V with respect to basis G for both the domain and codomain. (b) If Tn denotes the n-fold composition of T with itself, then show that An is the matrix of the linear transformation Tn : V V with respect to basis G for both the domain and codomain.