1. If A and B are m x n matricies, show that

U = [X in $\displaystyle R^n $ | AX = BX ] is a subspace of $\displaystyle R^n $

2. If [X1, X2, X3,......XK] is independant, show that [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] is also independant.

3.If [Y, X1, X2, X3,.....XK] is independant show that [Y + X1, Y + X2, Y + X3,....Y+XK] is also independant

4. Let U and W denote subspace $\displaystyle R^n $, and assume that U is a proper subset of W. If dimU = n-1, show that either W = U or W = $\displaystyle R^n $

5. Let U and W denote the subspaces of $\displaystyle R^n $, and assume that U is a proper subset of W. If dim W = 1, show that either U = {0} or U = W

6. Let C1, C2,....CN be fixed columns in $\displaystyle R^n $ and define T: $\displaystyle R^n $ ----> $\displaystyle R^m $ by $\displaystyle T([x1 *x2 ....... xN]^T = x1C1 + x2c2+.....+XnCn$ in $\displaystyle R^n $. Show that T is a linear transformation and find the matrix of T