Can you show us what you have tried? (Hint: For each linear transformation matrix, try reducing it to row echelon form and show us what you get. This will help in finding a basis for both matrices).
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
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 ∩ U2 and U1 + U2.
thank you in advance...