### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2213

 FINAL EXAMINATION December 1996 TIME: 3 HOURS

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

1. Let

(a) Find A-1 using row reduction.

(b) Find elementary matrices E1, E2, . . . , Ek (k is however many elementary matrices that you need) such that

Ek . . . E2E1A = I.

(c) Check your result by calculating Ek . . . E1 .

2. Let , and B = {b1, b2, b3}.

(a) Prove that B is a basis for . Quote any results you might use.

(b) Find [y]B .

(c) Find a matrix P such that P[x]B = x and a matrix Q such that [x]B = Q[x] for all x in .
(You may use part (c) to do part (b) if you wish.)

3. Let

(a) Find a basis for Nul A.

(b) Find a basis for Col A.

(c) Find a basis for Row A.

(d) What is the rank of A?

4. Find a basis for the subspace of spanned by

What is the dimension of this subspace?

5. For each of the following, determine the matrix A such that Tx = Ax .

(a) rotates vectors anticlockwise about the origin through an angle of radians.

(b) leaves the line y = x fixed and T(x, 0) = (2x, 0) for all x.

(c) and T(x1, x2, x3) = (3x2 - x3, x1 + 4x2 + x3).

6. Let

(a) Show that is an eigenvalue and find the corresponding eigenspace.

(b) Show that is an eigenvector for A and find the corresponding eigenvalue.

7. Let . It can be shown that the eigenvalues of A are 8 and -1 with corresponding eigenspaces

(a) What result allows you to immediately conclude that A is diagonalizable?

(b) Give a basis B for such that if Tx = Ax, then [Tx]B = D[x]B where D is a diagonal matrix.

(c) Produce a matrix P such that P-1AP is a diagonal matrix.

8. Which of the following sets are vector spaces? Give reasons whether the set is or is not a vector space. If it is a vector space, state the dimension.

(a) All polynomials of the form p(t) = 1 + at + bt2 (a,b can be any number).

(b) All polynomials p(t) in P5 whose derivative is 0 at t = 0.
(c) All points in the plane inside the unit circle (i.e. all points such that ).

9. (a) If the null-space of a 5 × 6 matrix is 3 dimensional, what are the dimensions of the row and column spaces?

(b) Let A be a 2 × 3 matrix. What are the possible dimensions of the null-space of A? Give an example of a 2 × 3 matrix A that has each of these possible dimensions.

10. (a) Prove that if is given by , then T is a linear transformation.

(b) If B = {1, t, t2, t3} find a matrix A such that [T(p(t))]B = A[p(t)]B. That is, find [T]B.

11. Suppose S = {v1, . . . ,vk+1} is a set of vectors in the vector space V, and suppose that vk+1 is a linear combination of S' = {v1, . . . ,vk}. Prove span{v1, . . . ,vk} = span{v1, . . . ,vk+1}.