# And working with the usual dot product of 3 use the

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and working with the usual dot product of 3 , use the Gram-Schmidt process to obtain an orthonormal basis for 3 . (b) Using your work from part (a), obtain a QR-factorization of the matrix 1 1 0 A = 1 0 1 0 1 1 .

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Final Exam/MAS3105 Page 6 of 7 5. (10 pts.) Let 1 0 0 A = 0 1 2 0 2 1 . (a) Compute the characteristic polynomial p A ( λ ) of A. Leave the polynomial in factored form. (Hint: Expand the required determinant using the first row or column. Don’t mess with the linear factor which is multiplied by the 2 x 2 determinant, which ends up being a difference of squares.) p A ( λ ) = (b) List the eigenvalues of A. (c) Obtain a basis, B λ , for each eigenspace, E λ , of A. Label correctly. (d) Obtain an invertible matrix P and a diagonal matrix D so that A = PDP -1 . P = D = (e) Verify your P and D work without finding the inverse of P. (Hint: There is a matrix equation that is equivalent to the one in part (d).
Final Exam/MAS3105 Page 7 of 7 6. (8 pts.) (a) Show that W = {a 0 + a 1 t + a 2 t 2 : a 0 + a 1 + a 2 = 0} is a subspace of P 2 .
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