(a) what do you mean by U\W? if you mean the set difference, this cannot possibly be a subspace of V, since since {0} is in the intersection, and thus NOT in U\W.

on the other hand, if you mean the quotient of U by W, this isn't necessarily even defined, since W may not be a subspace of U. something is missing, here...

(b) well first we need to determine the two kernels (solve two homogeneous sets of linear equations). for example:

ker(L_{1}) = {(x_{1},x_{2},x_{3},x_{4}) in R^{4}: L_{1}(x_{1},x_{2},x_{3},x_{4}) = (0,0)}

this is equivalent to the homogeneous system of linear equations:

3x_{1}+ x_{2}+ 2x_{3}- x_{4}= 0

2x_{1}+ 4x_{2}+ 5x_{3}- x_{4}= 0

or the matrix equation:

it should be clear that this matrix has rank 2, so the kernel has dimension 2.

row-reduction gives us:

so if we set x_{3}= s, x_{4}= t, we get:

10x_{1}= -3s + 3t

10x_{2}= -11s + t

so the kernel consists of vectors of the form (-3s+3t,-11s+t,10s,10t) = s(-3,-11,10,0) + t(3,1,0,10), so {(-3,-11,10,0),(3,1,0,10)} is a basis for U_{1}.

i leave it to you to construct a basis for U_{2}, and of U_{1}∩U_{2}, U_{1}+U_{2}.

you may find it helpful to know that dim(U_{1}∩U_{2}) ≤ min(dim(U_{1}),dim(U_{2})) and that:

dim(U_{1}+U_{2}) = dim(U_{1}) + dim(U_{2}) - dim(U_{1}∩U_{2}).