MA 401, Linear Algebra, Autumn 2007

Tutorial 2

1. Consider the following system of linear equations.

2x1 + 3x2 2x3 + 4x4 = 2

6x1 9x2 + 7x3 8x4 = 3

4x1 + 6x2 x3 + 20x4 = 13

Find the echelon form of the above system, row rank of the coefficient matrix A, a basis forthe null space of A and all the solutions.

2. let T : IR2 IR2 be defined by T (x1, x2) = (x2, x1). Find the matrix of T in the standardbasis as well as in the basis {[1, 2]t, [1,1]t}.

3. (a) Let {v1, v2, , vn be a basis for V and let T : V V be such that

Tvj = vj+1, j = 1, , n 1, Tun = 0.

Find the matrix representing T and show that T n = 0, but T n1 6= 0.

(b) Let S : V V be such that Sn = 0, but Sn1 6= 0. Prove that there exists a basis forV with respect to which the matrix of S is equal to the matrix of T in part (a).

4. Let T : U V, S : V W be linear transformations. Show that

rank (ST ) min{rank(S), rank(T )}.

Find sufficient conditions on T and S so that

i) rank(ST ) = rank(S), ii) rank(ST ) = rank(T ).

5. Let T : V V be such that rank (T 2) = rank (T ). Show that R(T )

N(T ) = {0}.

6. Let T : IR3 IR2 and S : IR2 IR3 be linear transformations. Show that ST is notinvertible.