### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2213

 FINAL EXAMINATION APRIL 1999 TIME: 3 HOURS

ANSWER ALL QUESTIONS WITHOUT USE OF A CALCULATOR. CREDIT WILL BE GIVEN FOR METHOD AND PRESENTATION OF SOLUTIONS.

 MARKS (8) 1. Row reduce the augmented matrix, to an echelon form and, hence, determine all solutions of the linear system Ax = b. (8) 2. Let (a) Compute detA. (b) Use elementary row operations to determine the inverse of A. (c) Factor A as a product LU, wherein L denotes a lower triangular matrix with unit diagonal entries and U denotes an upper triangular matrix. (9) 3. (a) Given and , determine whether or not the vector belongs to the subspace, span , in . (b) Find a basis for the column space of the matrix, Justify your answer. (c) Let be three non-zero vectors from . Prove that if the vectors are mutually orthogonal then they are linearly independent. (10) 4. (a) Let and be two bases of in which (i) Determine the change of coordinates matrix, P, for the change to . (ii) Given the coordinates, , for a vector relative to , find . (b) Determine an orthogonal base for the subspace in which (10) 5. (a) Let be a linear mapping for which where {e1, e2, e3} is the standard basis for . (i) Is T a one-to-one mapping? (ii)   Is T an onto mapping? Justify your answers to these questions. (b) Let be the linear transformation for which . If where calculate the matrix, , for T relative to base . (8) 6. Find the eigenvalues and eigenvectors of the matrix . (10) 7. The vectors constitute a linearly independent set of eigenvectors of the symmetric matrix, (a) Verify that the eigenvectors are mutually orthogonal. (b) Construct an orthogonal matrix P with which A is orthogonally diagonalizable as A = PDP-1 for some diagonal matrix D. (c) Find (i) the matrix P-1, and (ii) the matrix D. (70)