Linear Algebra

\f\f3) (20 Points) Let S be the ]inear transformation from the vector space R2 to R2 itself given by x1 3x1 + x2 5( )= X2 XI + 3x2 (a) Verify that the vectors a] M] are eigenvectors of the linear transformation 5, both by nding them and showing they satisfy the definition of Eigenvector. Conclude that B' = {v1,v2} is a basis of R2 consisting of eigenvectors. (b) Find the matrix M = [5]]3,l of S with respect to the basis 8' = {v1,v2}. For the following parts, let P1 be the vector space of all real polynomials of degree 1 or less. Consider the linear transformation T:P1)P1 dened by T(a + I?!) = a +317 + (311 + b)! for any (Ii-b! E P1. (c) Find the image of f( t) = 3 + 5! under the linear transformation T. ((1) With respect to the basis B={I, I}, nd the matrix A of the linear transformation T relative to B. A = [T]B (e) Find a basis C of the vector space P1 such that the matrix of T with respect to C is a diagonal matrix. (f) Express K!) = 3 + 5! as a linear combination of basis vectors of C. (3) Forf) = 3 + 5! , nd the coordinate vectors for mills, [T(f(t))]B, [tllB's [T(f{t))]B', [Efc- [thtlllic (h) Check that M[f(r)]B' = [roam];- (i) Fill in the mapping diagram below. \f