Linear Algebra Problem involving subspaces and subsets?

Question

Given subspaces H and K of a vector space V, the sum of H and K, written as H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K; that is,
H + K = {w : w = u + v for some u in H and some v in K }

a. Show that H + K is a subspace of V.
b. Show that H is a subspace of H + K and K is a subspace of H+ K.

I am completely lost on this problem so any information on a solution or how to solve it would be greatly appreciate. Thanks a lot.

Answer 1

Since you already know that H and K come from a set which is known to be a vector space, your job is easy (for both problems). You must verify that 0 is an element of H + K, and also that H + K is closed under the taking of linear combinations. That is, if h1 + k1 and h2 + k2 are two elements of H + K, show that c1(h1 + k1) + c2(h2 + k2) is also in H + K (which it is, of course). Do the same for the second question.

Answer 2

For part b, I would prefer an argument like this:

Suppose h is in H. Then h = h + 0 is in H + K, so H is a subset of H + K, so is a subspace of H + K.

Answer 3

Ok you have the right properties to prove, but you are off on the wrong foot. Let w be an element of W. Then w = (s, s-t,t) for some real numbers s and t. Let u be another element of W. Then u = (s',s'-t',t') for some real numbers s' and t'. u+v = (s+s', s-t + s'-t' , t + t') = (s+s', s+s' - (t+t'), t+t'). Hence u+v is an element of W with the choice of real numbers: s+s' and t+t'. Study this argument carefully and I'm sure you can show the other properties. good luck