ID. .LetB=( 2 1 . Consider the matrix A = (1 2). Diagonalize the matrix if possible. 1 1 0 1). Determine 32, B3, and generalize for B". 7 1 . Suppose C: (2 3 ). (a) Find the eigenvalues and their corresponding eigenvectors. (b) Use your result to diagonalize C if possible. (c) If C is diagonalizable, what are C2 and C3? 1 0 . Let V be the subspace of R4 spanned by [3 and [1) . Find an orthonormal basis for V and for D 1 VJ: 1 . Let V be the subspace of R3 spanned by 2 . What is the orthogonal complement of V? 3 1 2 . Let A be a matrix such that A = 3 4 . Find AT and (ATP. 5 6 . 1 2 5 6 . . . Let A and B be rnatrlces such that A = 3 4 and B = 7 8 . Prove or disprove the follow1ng statement: (A + B)T = AT + .31". (a) Give an example of an orthogonal basis for R2. 2 3 onto the subspace spanned by your basis. (b) Given the vector [ J and the orthogonal basis from part (a), nd the projection of this vector . For the following vectorsI determine whether they form an orthogonal set: 1 2 (a) 0 , 2 1 2 1 1 0 (b) 1 , 2 , 0 1 1 3 Consider the vector space P202) of all realvalued polynomials of degree at most 2. We say that two polynomials 10(3) and 9(3) are orthogonal if their inner product, dened as If] p(.r)q(:r)dx, equals zero. (a) Verify that Mac) = 1 and 9(3) = :12 are orthogonal. (b) Find a polynomial 1(3) that is orthogonal to both 30(3) and (x)