1. Let B = (b1, b2) and C = (c1, C2) be bases for a vector space V, and suppose that b, = - c1 + 4c2 and b2 = 501 - 3c2, a) Find the change of basis matrix from B to C. b) Find [x]c for x = 5b, + 3b2. 2. Let B = (b,, b,} and C = (c1, C2) be bases for R2, with a) Find the change of basis matrix from B to C. b) Find the change of basis matrix from C to B. 3. In P2, find the change of basis matrix from the basis B = {1 - 312, 2 + t -5t2, 1+ 2t} to the standard basis {1, t, t'}. Then write t as a linear combination of the polynomials in B. 1. Determine if A = - 2 is an eigenvalue of 3 3, 5. Determine if -1+ v2| is an eigenvector or If so, find the eigenvalue. 3 7 6. Determine if is an eigenvector of . Is so, find the eigenvalue. 2 -5 4 7. In the following problems, find a basis for the eigenspace corresponding to each listed eigenvalue for the matrix a) 1 = 1 b) 1 =54 0 O 8. Find the eigenvalues of the matrix [0 0 0 l . 1 0 3 9. Show that if A2 is the zero matrix, then the only eigenvalue of A must be zero. 10. Show that A is an eigenvalue of A if and only if :1 is an eigenvalue of AT. (Hint: nd out how/l M andAr ll are related) 11. Find the characteristic polynomial and the eigenvalues of the following matrices. 5 3 a) ls l 3 4 b ) l4 s l 12. It can be shown that the algebraic multiplicity of an eigenvalue .1 is always greater than or equal to the dimension of the eigenspaee corresponding to A. Find it in the matrix A below such that the eigenspaee for A : 5 is twodimensional: 13. Find the characteristic polynomials and eigenvalues (including their multiplicities) of the following matriees: 5 -2 3 a) 0 l 0 6 7 2 5 U 0 0 8 4 0 0 b) '0 7 l 0 l 5 2 1 (Hint: Don't try to use new reduction to compute these determinants, nding the characteristic polynomial will be difficult this way because there are is oating around)