Final-ex

# Use the gram schmidt process to obtain an orthonormal

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, 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 A= .

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Final Exam/MAS3105 Page 6 of 7 5. (10 pts.) Let 100 A= 012 021 . (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 the2x2 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 ,o fA . 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 1 2 =0 } is a subspace of P 2 . (b) Obtain a basis for W. (Hint: Use the coordinate mapping with respect to the natural basis for P 2 to obtain an equivalent problem in 3 . Solve that problem and translate the results to the polynomial space. Observe that the linear equation you have to solve is the one giving the defining condition on the members of W.) 7. (7 pts.) Let T: P 2 P 2 be the function defined by the rule T(p) = p -2 p , where the primes denote differentiation with respect to t. (a) Verify that T is a linear transformation. (You need only quote the appropriate properties of differentiation.) (b) Obtain the matrix [T] B , whereB={ 1 ,1+t ,1+t+t 2 }i s an off the wall basis for P 2 . (c) What can you tell about the dimensions of the kernel and the range by using the matrix [T] B ??
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