be a basis for a vector space V and let be a linear

Problem 32E [M] Let T be the transformation whose standard matrix is given below. Find a basis for R4 with the property that is diagonal.

Problem 3E be the standard basis for R3, let be a basis for a vector space V , and let be a linear transformation with the property that

Problem 1E Let be bases for vector spaces V and W , respectively. be a linear transformation with the property that Find the matrix for T relative to B and D.

Problem 2E Let D = {d1,d2} and B = {b1, b2} be bases for vector spaces V and W , respectively. Let be a linear transformation with the property that Find the matrix for T relative to D and B.

Problem 4E be a basis for a vector space V and let be a linear transformation with the property that Find the matrix for T relative to B and the standard basis for R2.

Problem 5E be the transformation that maps a polynomial p(t) into the polynomial a. Find the image of b. Show that T is a linear transformation. c. Find the matrix for T relative to the bases

Problem 7E Assume the mapping defined by is linear. Find the matrix representation of T relative to the basis

Problem 6E be the transformation that maps a polynomial p(t) into the polynomial a. Find the image of b. Show that T is a linear transformation. c. Find the matrix for T relative to the bases

Problem 8E be a basis for a vector space V . Find when T is a linear transformation from V to V whose matrix relative to B is

Problem 9E a. Find the image under b. Show that T is a linear transformation. c. Find the matrix for T relative to the basis and the standard basis for R3.

Problem 10E a. Show that T is a linear transformation. b. Find the matrix for T relative to the basis for P3 and the standard basis for R4.

Problem 11E In Exercises 11 and 12, find the B-matrix for the transformation

Problem 13E In Exercises 13–16, define Find a basis B for R2 with the property that is diagonal.

Problem 12E In Exercises 11 and 12, find the B-matrix for the transformation

Problem 14E In Exercises 13–16, define Find a basis B for R2 with the property that is diagonal.

Problem 16E In Exercises 13–16, define Find a basis B for R2 with the property that is diagonal.

Problem 15E In Exercises 13–16, define Find a basis B for R2 with the property that is diagonal.

Problem 17E a. Verify that b1 is an eigenvector of A but that A is not diagonalizable. b. Find the B-matrix for T .

Problem 18E matrix with eigenvalues 5, 5, and –2. Does there exist a basis B for R3 such that the B-matrix for T is a diagonal matrix? Discuss.

Problem 20E Verify the statements in Exercises 19–24. The matrices are square. If A is similar to B, then A2 is similar to B2.

Problem 21E Verify the statements in Exercises 19–24. The matrices are square. If B is similar to A and C is similar to A, then B is similar to C.

Problem 19E Verify the statements in Exercises 19–24. The matrices are square. If A is invertible and similar to B, then B is invertible and A–1 is similar to B–1. [Hint: for some invertible P. Explain why B is invertible. Then find an invertible Q such that

Problem 22E Verify the statements in Exercises 19–24. The matrices are square. If A is diagonalizable and B is similar to A, then B is also diagonalizable.

Problem 23E Verify the statements in Exercises 19–24. The matrices are square. If B D P–1AP and x is an eigenvector of A corresponding to an eigenvalue , then P–1x is an eigenvector of B corresponding also to .

Problem 24E Verify the statements in Exercises 19–24. The matrices are square. If A and B are similar, then they have the same rank. [Hint: Refer to Supplementary Exercises 13 and 14 in Chapter 4.] Reference Exercises 13 in Chapter 4 Develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n. Show that if P is an invertible m × m matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and P–1 (PA)] Exercise 12: Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.) a. Show that if B is n × p, then rank AB rank A. [Hint: Explain why every vector in the column space of AB is in the column space of A.] b. Show that if B is n × p, then rank AB rank B. [Hint: Use part (a) to study rank.(AB)T .] Reference Exercises 14 in Chapter 4 Develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n. Show that if Q is invertible, then rank AQ = rank A. [Hint: Use Exercise 13 to study rank (AQ)T .] Exercise 13: Show that if P is an invertible m × m matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and P–1 (PA)] Exercise 12: Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.) a. Show that if B is n × p, then rank AB rank A. [Hint: Explain why every vector in the column space of AB is in the column space of A.] b. Show that if B is n × p, then rank AB rank B. [Hint: Use part (a) to study rank.(AB)T .]

Problem 25E The trace of a square matrix A is the sum of the diagonal entries in A and is denoted by trA. It can be verified that for any two n × n matrices F and G. Show that if A and B are similar, then tr A = tr B.

Problem 27E Let V be with the standard basis, denoted here by ; and consider the identity transformation . Find the matrix for I relative to . What was this matrix called in Section 4.4?

Problem 26E It can be shown that the trace of a matrix A equals the sum of the eigenvalues of A. Verify this statement for the case when A is diagonalizable.

Problem 28E Let V be a vector space with a basis , let W be the same space V with a basis , and let I be the identity transformation I W V ! W . Find the matrix for I relative to B and C. What was this matrix called in Section 4.7?

Problem 29E Let V be a vector space with a basis . Find the B-matrix for the identity transformation

Problem 30E [M] In Exercises 30 and 31, find the B-matrix for the transformation

Problem 31E [M] In Exercises 30 and 31, find the B-matrix for the transformation