Mathematics 700 Test 1

1. (20 points) Let T : R^{2} → R^{2} be the linear map T (x, y) = (2x − 3y, x − 2y).

(a) Find the matrix of T in the standard basis.

(b) Find the matrix of T in the ordered basis (0, 3), (1, −1).

(c) Find the inverse of T. (The answer should be expressed in the same form as T and not as a matrix.)

(d) Find a basis for both the range and null space of the linear transformation T − I.

(e) Find T^{2}.

(f) Find T^{1995}

2. (10 points) Give an example of linear maps T : R^{2} → R^{4} and S : R^{4} → R^{2} so that ST is invertible.

3. (10 points) Let T : R^{4} → R^{2} and S : R^{2} → R^{4} be linear maps. Show nullity(ST) ≥ 2.

4. (10 points) If u, v, w are linearly independent vectors in the vector space V show that u, u + v and u + v + w are also linearly independent.

5. (10 points) Let v_{1} = (1, 0, 0), v_{2} = (2, 1, 0), v_{3} = (3, 2, 1). Then v_{1}, v_{2}, v_{3} is a basis of R^{3}. Find the basis of R^{3∗} dual to v_{1}, v_{2}, v_{3}.

6. (10 points) Let U and W be subspaces of the finite dimensional vector space V. Assume that dim U + dim W = dim V and U + W = V. Then show V = U ⊕ W.

7. (10 points) Let C = {x + iy : x, y ∈ R} be the complex numbers. Then C is a two dimensional vector space over the real numbers R. Let T : C → C be multiplication by a + bi, that is T z = (a + bi)z Then find the matrix of T in basis 1, i of C.

8. (10 points) Find explicitly a linear map S : R^{2} → R^{3} so that S(2, 3) = (1, 2, 3) and S(1, 2) = (4, 5, 6).

9. (10 points) Gives examples of

(a) Matrices A and B with AB ≠ BA.

(b) A matrix with A^{3} = 0 but A^{2} ≠ 0.

Mathematics 700 Test 1

This is to be done in three hours in one setting. I would prefer that it is closed book, but if you feel that you have to look up a something write me a note like “I looked up the definition of rank and used it in problems numbers 2 and 5”. I will then take a little off on these problems, say 20%. This is due in class next Tuesday. Good luck.