Number 17 (p) =entriesand isp(1)p(0)a. Show that T is a linear transformation.tr(FG) = tr(GF) for any two n x n mShow that if A and B are similar, then trb. Find the matrix for T relative to the basis {1, t, t²,t³} forP3 and the standard basis for R“.26. It can be shown that the trace of a matrixthe eigenvalues of A. Verify this statemerA is diagonalizable.In Exercises 11 and 12, find the B-matrix for the transformationX Ax, where B = {b1, b2}.27. Let V be R" with a basis B = {b1,...with the standard basis, denoted here byidentity transformation I : R" → R", wEthe matrix for I relative to B and E. Wcalled in Section 4.4?-4 -111. A =Di =b2 =-628. Let V be a vector space with a basis B =be the same space V with a basis C = {be the identity transformation I :V →for I relative to B and C. What was12. A =b, =b, =In Exercises 13–16, define T : R² → R² by T(x) = Ax. Find abasis B for R2 with the property that [T]B is diagonal.Section 4.7?13. А %3D14. A =29. Let V be a vector space with a basis Bthe B-matrix for the identity transform=416. A =15. A =3[M] In Exercises 30 and 31, find the B-mattion x→ Ax where B = {b,, b2, b3}.17. Let A =and B = {b1, b2}, for bi =-126 -2 -230. A = | 3 1 -2b, =Define T : R2 → R² by T(x) = Ax.2 -2[1] 2a. Verify that bị is an eigenvector of A but that A is notdiagonalizable.[-1b3 =b1 =1 , b2 =1-1b. Find the B-matrix for T.18. Define T :R³ → R³ by T(x) = Ax, where A is a 3 x 3matrix with eigenvalues 5, 5, and -2. Does there exist a basisB for R3 such that the B-matrix for T is a diagonal matrix?T-7 -48 –16131. A =146.-3 -45 -19Discuss.-3Verify the statements in Exercises 19-24. The matrices are square.b =1, b2 =1, b3 =-319. If A is invertible and similar to B, then B is invertibleand A-1 is similar to B-1. [Hint: P-AP = B for some32. [M] Let T be the transformation wgiven below. Find a basis for R4 withis diagonal.invertible P. Explain why B is invertible. Then find aninvertible Q such that Q-A-Q = B¬1.]-649.20. If A is similar to B, then A² is similar to B2.-30.16.21. If B is similar to A and C is similar to A, then B is similarA =-1-210.to C.-440.7

Name: Number 17 (p) =entriesand isp(1)p(0)a. Show that T is a linear transfo
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Description: Number 17 (p) =entriesand isp(1)p(0)a. Show that T is a linear transformation.tr(FG) = tr(GF) for any two n x n mShow that if A and B are similar, then trb. Find the matrix for T relative to the basis {1, t, t²,t³} forP3 and the standard basis for R“

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Number 17 (p) =entriesand isp(1)p(0)a. Show that T is a linear transformation.tr(FG) = tr(GF) for any two n x n mShow that if A and B are similar, then trb. Find the matrix for T relative to the basis {1, t, t²,t³} forP3 and the standard basis for R“.26. It can be shown that the trace of a matrixthe eigenvalues of A. Verify this statemerA is diagonalizable.In Exercises 11 and 12, find the B-matrix for the transformationX> Ax, where B = {b1, b2}.27. Let V be R&#34; with a basis B = {b1,...with the standard basis, denoted here byidentity transformation I : R&#34; → R&#34;, wEthe matrix for I relative to B and E. Wcalled in Section 4.4?-4 -111. A =Di =b2 =-628. Let V be a vector space with a basis B =be the same space V with a basis C = {be the identity transformation I :V →for I relative to B and C. What was12. A =b, =b, =In Exercises 13–16, define T : R² → R² by T(x) = Ax. Find abasis B for R2 with the property that [T]B is diagonal.Section 4.7?13. А %3D14. A =29. Let V be a vector space with a basis Bthe B-matrix for the identity transform=416. A =15. A =3[M] In Exercises 30 and 31, find the B-mattion x→ Ax where B = {b,, b2, b3}.17. Let A =and B = {b1, b2}, for bi =-126 -2 -230. A = | 3 1 -2b, =Define T : R2 → R² by T(x) = Ax.2 -2[1] 2a. Verify that bị is an eigenvector of A but that A is notdiagonalizable.[-1b3 =b1 =1 , b2 =1-1b. Find the B-matrix for T.18. Define T :R³ → R³ by T(x) = Ax, where A is a 3 x 3matrix with eigenvalues 5, 5, and -2. Does there exist a basisB for R3 such that the B-matrix for T is a diagonal matrix?T-7 -48 –16131. A =146.-3 -45 -19Discuss.-3Verify the statements in Exercises 19-24. The matrices are square.b =1, b2 =1, b3 =-319. If A is invertible and similar to B, then B is invertibleand A-1 is similar to B-1. [Hint: P-AP = B for some32. [M] Let T be the transformation wgiven below. Find a basis for R4 withis diagonal.invertible P. Explain why B is invertible. Then find aninvertible Q such that Q-A-Q = B¬1.]-649.20. If A is similar to B, then A² is similar to B2.-30.16.21. If B is similar to A and C is similar to A, then B is similarA =-1-210.to C.-440.7

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Let us consider a n x n matrix A  then the eigen value and eigen vector is determined as

Determine…

4 17. Let A = and B = {b1, b2}, for b = b2 = Define T: R? -→ R² by T(x) = Ax. a. Verify that b, is an eigenvector of A but that A is not diagonalizable. [-]- b. Find the B-matrix for T.