help me those hw questions

If a problem asks you to justify your answer, you need to write a few words explaining your conclusion. You may not use a calculator on this exam. 1. Consider A= TI. v. - a. Verify that v, and v, are eigenvectors of A. b. Diagonalize A. c. Compute As. 2 0 -2 2. The eigenvalues of A= 1 3 2 , including repetition, are 1 = 2 and 2 = 3. Diagonalize 0 0 3 A [just find P and D, don't worry about P'], or explain why it can't be diagonalized. -3 3. Let m = -1 and w = -8 8 a. Find m b. Find a unit vector in the direction of m . c. Evaluate m . w. 4. If x is orthogonal to u and v, show that x is orthogonal to u + v. 5. Consider the vectors v, = 1 and v 2 = -61 2 1 , which are orthogonal vectors. a. Express X = as a linear combination of v, and V2 . b. Use v, and v2 to produce an orthonormal basis for 2 . 6. Consider the vectors u, = and u, = 3 , let W be the subspace spanned by u, and u2 . a. Verify that u, and u2 are orthogonal. b. Find the point in W that is closest to y = A c. Write y from part (b) as the sum of a vector in W and a vector orthogonal to W.5 7. The vectors and 6 form a basis for a subspace W. Use the Gram-Schmidt process to find an orthogonal basis for W. 8. Consider the inner product space C [0,1] with the inner product defined as ( f.8 ) = [ f(1)8 (1) dt. Compute | f (t)|, where f ( 1 ) = P . 3 1 9. Consider A= 1 3 0 0 0 2 a. Explain why A is orthogonally diagonalizable. b. Find the orthogonal diagonalization of A, given that the characteristic equation is (2-a) (4-2) = 0. [Just find P and D, don't worry about P.]