8.4 Write on paper for upvote please. Questions: 12, 17, 22

8.4 Matrices for General Linear Transformations 485 b. Verify that Formula (8) holds for every vector in R'. d. Use the formula obtained in (c) to compute T CONN C. Is T one-to-one? If so, find the matrix of T- with respect to the basis B. 7. Let T : P2 - P2 be the linear operator T(p(x)) = p(2x + 1), that is, 11. Let A = 2 0 s be the matrix for T : P 2 - P 2 with I ( co + qx + cax ? ) = co + 9 ( 2x + 1) + c2 (2x + 1)2 respect to the basis B = {V,, V2, V3}, where v, = 3x + 3x2, a. Find [T]B with respect to the basis B = {1, x, x2]. V2 = -1+3x+2x2, v3=3+7x+2x2. b. Use the three-step procedure illustrated in Example 2 to a. Find [T(v,)IB, [T(v2)]B, and [T(v,)JB. compute T(2 - 3x + 4x2). b. Find T(v,), T(v2), and T(v;). c. Check the result obtained in part (b) by computing T(2 - 3x + 4x2) directly. c. Find a formula for T(a, + ax + azx2 ). 8. Let T : P2 - P3 be the linear transformation defined by d. Use the formula obtained in (c) to compute T(1 + x2). T(p(x)) = xp(x - 3), that is, 12. Let T1 : P1 - P2 be the linear transformation defined by T ( Co + Cix + (2x2 ) = x(co + c (x-3)+ c2(x-3)2) Ti (p(x)) = xp(x) a. Find [T ]B',B relative to the bases B = {1, x, x2} and and let T2 : P2 - P2 be the linear operator defined by B' = {1, x, x , x ]. T2 (p(x) ) = p(2x + 1) b. Use the three-step procedure illustrated in Example 2 to compute T (1 + x - x2). Let B = {1, x} and B' = {1, x, x2) be the standard bases for P, and P2. . Check the result obtained in part (b) by computing T(1 + x - x2) directly. a. Find [T2 . TilB', B, [Tz]B', and [T,]B', B. b. State a formula relating the matrices in part (a). Let v1 = 3 and v2 = ]. and let A = [_ _ ] be the . Verify that the matrices in part (a) satisfy the formula you matrix for T : R2 - R2 relative to the basis B = {v1, V2). stated in part (b). a. Find [T(v, )]B and [T(v2) ]B. 13. Let T1 : P1 - P2 be the linear transformation defined by b. Find T(v, ) and T(v2). Ti(Co + C1x) = 200 - 30,x c. Find a formula for T (x ]) and let T2 : P2 - P3 be the linear transformation defined by d. Use the formula obtained in (c) to compute T(). Let B = {1, x], B" = {1,x, x ], and B' = {1,x,x?, x3). a. Find [T2 . TilB', B, [T2]B', B" , and [T, 1B", B. b. State a formula relating the matrices in part (a). 10. Let A = 1 6 2 1 be the matrix for T : R* - R3 rel- c. Verify that the matrices in part (a) satisfy the formula you stated in part (b). ative to the bases B = {V1, V2, V3, VA} and B' = {w, w2, Wal, where 14. Let B = {V1, V2, V3, VAJ be a basis for a vector space V. Find the matrix with respect to B for the linear operator T : V - V defined by T(v1 ) = V2, T(V2) = V3, T(V3) = VA, T(v.) = v1. 15. Let T : P2 - M22 be the linear transformation defined by T(P) = [P(0) P(1)] p(-1) p(0)] W. = 8. W. = =[ let B be the standard basis for M2, and let B' = {1, x, x2}, B" = {1, 1 + x, 1 + x} be bases for P2. a. Find [T(v,)]B', [T(v2)]B' , [T(v;)]B', and [T(v4)IB. a. Find [T] B, B and [T]B, B". b. Find T(v,), T(v2), T(v;), and T(v4). b. For the matrices obtained in part (a), find T(2 + 2x+ x2 ) c. Find a formula for T * 2 using the three-step procedure illustrated in Example 2. X 3 . Check the results obtained in part (b) by computing T(2 + 2x + x2) directly. 486 CHAPTER 8 General Linear Transformations 16. Let T : M22 - R2 be the linear transformation given by Direct computation (x) ( [ a a] ) = [ a + 6 + q] (1) (3 Multiply by s'.B (T(x)JB and let B be the standard basis for M22, B' the standard basis ( 2 ) for R2, and FIGURE Ex-20 a. Find [T]B'. B and [T]B", B. 21. In each part, fill in the missing part of the equation. b. Compute T (3 )using the three-step procedure that a. [T2 . TilB', B = [T2] ? [T , IB", B was illustrated in Example 2 for both matrices found in b. [T3 . T2 0 TilB',B = [T;] ? [T, IBM , B" [T , IB", B part (a). c. Check the results obtained in part (b) by computing Working with Proofs T (3 2 ) directly. 22. Prove: If T : V - W is the zero transformation, then the matrix for T with respect to any bases for V and W is a zero matrix. 17. (Calculus required) Let D : P2 - P2 be the differentiation 23. Prove: If B and B' are the standard bases for R" and Rm, operator D(p) = p'(x). respectively, then the matrix for a linear transformation T : R" - R' relative to the bases B and B' is the standard a. Find the matrix for D relative to the basis B = {P1, P2, P3} matrix for T. for P2 in which p, = 1, P2 = x, P3 = x2. b. Use the matrix in part (a) to compute D(6 - 6x + 24x2). True-False Exercises TF. In parts (a)-(e) determine whether the statement is true or 18. (Calculus required) Let D : P2 - P2 be the differentiation false, and justify your answer. operator D(P) = P' (x). a. If the matrix of a linear transformation T : V - W rel- a. Find the matrix for D relative to the basis B = {P1 , P2, P3 for P2 in which p, = 2, P2 = 2 - 3x, P3 = 2- 3x+ 8x2. ative to some bases of V and W is 3, then there is a b. Use the matrix in part (a) to compute D(6 - 6x + 24x2). nonzero vector x in V such that T(x) = 2x. b. If the matrix of a linear transformation T : V - W rel- 19. (Calculus required) Let V be the vector space of real-valued functions defined on the interval (-oo, co), and let ative to bases for V and W is 3 . then there is a D : V - V be the differentiation operator. nonzero vector x in V such that T(x) = 4x. . Find the matrix for D relative to the basis B = {f,, f2, f} } for V in which f, = 1, f2 = sinx, f; = cosx. c. If the matrix of a linear transformation T : V - W rel- ative to certain bases for V and W is 2 3 , then T is b. Use the matrix in part (a) to compute one-to-one. D(2 + 3 sinx - 4 cosx) d. If S : V - V and T : V - V are linear operators and B 20. Let V be a four-dimensional vector space with basis B, let is a basis for V, then the matrix of S . T relative to B is W be a seven-dimensional vector space with basis B', and let [TIBISIB. T : V - W be a linear transformation. Identify the four vec- e. If T : V - V is an invertible linear operator and B is tor spaces that contain the vectors at the corners of the accom- a basis for V, then the matrix for T- relative to B is panying diagram. [T ]B