(a) Suppose that

V is vector spaces over a field F and that U and W are subspaces of V .

Show that

U \W is also a subspace of V

(b) Define a real linear transformation

L1 : R4 -->R2 by

L1(x1, x2, x3, x4) = (3x1 + x2 + 2x3 − x4, 2x1 + 4x2 + 5x3 − x4)

and let

U1 denote the kernel of L1. Define a real linear transformation L2 : R4 --> R2 by

L

L2=(x1, x2, x3, x4) = (5x1 + 7x2 + 11x3 + 3x4, 2x1 + 6x2 + 9x3 + 4x4)

and let

U2 denote the kernel of L2. Construct bases for U1, U2, U1 nU2 and U1 + U2.