Problem 1PP: Find T(a0 + a1t + a1t2), if T is the linear transformation from 2 to 2 whose matrix relative to B =... Problem 2PP: Let A, B, and C be n n matrices. The text has shown that if A is similar to B, then B is similar to... Problem 1E: Let B = b1,b2,b3 and D = d1,d2 be bases for vector spaces V and W, respectively. Let T : V W be a... Problem 2E: Let D = {d1, d2} and B = {b1, b2} be bases for vector spaces V and W, respectively. Let T : V W be... Problem 3E: Let = e1,e2,e3 be the standard basis for 3, B = b1,b2,b3 be a basis for a vector space V, and T : 3... Problem 4E: Let B = b1,b2,b3 be a basis for a vector space V and T : V 2 be a linear transformation with the... Problem 5E: Let T : 2 3 be the transformation that maps a polynomial p(t) into the polynomial (t + 5)p(t). a.... Problem 6E: Let T : 2 4 be the transformation that maps a polynomial p(t) into the polynomial p(t) + t2p(t). a.... Problem 7E: Assume the mapping T : 2 2 defined by T(a0 + a1t +a2t2) = 3a0 + (5a0 2a1)t + (4a1 + a2)t2 is... Problem 8E: Let B = {b1, b2, b3} be a basis for a vector space V. Find T (3b1 4b2) when T is a linear... Problem 9E: Define T :2 3 = by T (p) = [p(-1)p(0)p(1)]. a. Find the image under T of p(t) = 5 + 3t. b. Show... Problem 10E: Define T : 3 4 by T(p) = [p(-3)p(-1)p(1)p(3)]. a. Show that T is a linear transformation. b. Find... Problem 11E: In Exercises 11 and 12, find the B-matrix for the transformation x Ax, when B = {b1, b2}. 11. A =... Problem 12E: In Exercises 11 and 12, find the B-matrix for the transformation x Ax, when B = {b1, b2}. 12. A =... Problem 13E: In Exercises 1316, define T : 2 2 by T(x) = Ax. Find a basis B for 2 with the property that [T]B is... Problem 14E: In Exercises 1316, define T : 2 2 by T(x) = Ax. Find a basis B for 2 with the property that [T]B is... Problem 15E: In Exercises 1316, define T : 2 2 by T(x) = Ax. Find a basis B for 2 with the property that [T]B is... Problem 16E: In Exercises 1316, define T : 2 2 by T(x) = Ax. Find a basis B for 2 with the property that [T]B is... Problem 17E: Let A = [1113] and B = {b1, b2}, for b1 = [11], b2 = [54]. Define T : 2 2 by T(x) = Ax. a. Verify... Problem 18E: Define T : 3 3 by T (x) = Ax, where A is a 3 3 matrix with eigenvalues 5 and 2. Does there exist a... Problem 19E: Verify the statements in Exercises 1924. The matrices are square. 19. If A is invertible and similar... Problem 20E: Verify the statements in Exercises 1924. The matrices are square. 20. If A is similar to B, then A2... Problem 21E: Verify the statements in Exercises 1924. The matrices are square. 21. If B is similar to A and C is... Problem 22E: Verify the statements in Exercises 1924. The matrices are square. 22. If A is diagonalizable and B... Problem 23E: Verify the statements in Exercises 1924. The matrices are square. 23. If B = P1AP and x is an... Problem 24E: Verify the statements in Exercises 1924. The matrices are square. 24. If A and B are similar, then... Problem 25E: The trace of a square matrix A is the sum of the diagonal entries in A and is denoted by tr A. It... Problem 26E: It can be shown that the trace of a matrix A equals the sum of the eigenvalues of A. Verify this... Problem 27E: Let V be n with a basis B = {b1 ,, bn}; let W be n with the standard basis, denoted here by and... Problem 28E: Let V be a vector space with a basis B = {b1, , bn}, W be the same space as V with a basis C = {c1, ... Problem 29E: Let V be a vector space with a basis B = {b1, bn}. Find the B-matrix for the identity... Problem 30E: [M] In Exercises 30 and 31, find the B-matrix for the transformation x Ax when B = {b1, b2, b3}.... Problem 31E: [M] In Exercises 30 and 31, find the B-matrix for the transformation x Ax when B = {b1, b2, b3}.... format_list_bulleted