### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2213

 FINAL EXAMINATION December 1997 TIME: 3 HOURS

SHOW ALL WORK. YOUR METHOD IS MORE IMPORTANT THAN THE RIGHT ANSWER.

 MARKS 1. Let (2) (a) Find a basis for Nul A. (2) (b) Find a basis for Col A. (2) (c) Find a basis for Row A. (2) (d) What is the rank of A? 2. Let , and . (3) (a) Prove that B is a basis for . Quote any results you might use. (2) (b) Find . (3) (c) Find a matrix P such that and a matrix Q such that for all in . (You may use part (c) to do part (b) if you wish.) 3. Let (2) (a) Find elementary matrices (k is the number of elementary matrices that you need) such that where U is upper triangular. (2) (b) Find a lower triangular matrix L, such that A = LU. (U as in part (a)). (4) 4. Why is the following set a subspace of ? Find a basis for it. (4) 5. Let Show that is an eigenvalue for A and find a basis for the corresponding eigenspace. (4) 6. Orthogonally diagonalize the matrix What easily checked property of this matrix guarantees that it is orthogonally diagonalizable? (4) 7. Find an orthonormal basis for the subspace of spanned by (4) 8. Let Show that is not in Col (A) and find a least squares solution of . (4) 9. Let and be bases for a 2 dimensional subspace of . Suppose and . Find the change of coordinate matrices and . (4) 10. Let T be a linear transformation from onto . If are independent vectors in , prove that are also independent vectors in . (2) 11. Let A be a matrix. If null(A) has dimension 2, what are the dimensions of the row and column spaces of A? (4) 12. Suppose is a solution to the system of equations (i.e. ). Prove that any solution to can be written where is a solution to . 13. A is an m by n matrix of rank r. Suppose has no solution for some right sides and infinitely many solutions for some other right sides . (2) (a) Decide whether the nullspace of A contains only the zero vector and why. (2) (b) Decide whether the column space of A is all of and why. (2) (c) Can there be a right side for which has exactly one solution? Why or why not? (60)