I'm still trying to understand subspaces. Here is my problem.
Given subspaces $H$ and $K$ of a vector space $V$, the sum of H and K, written $H + K$, 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}.
i. Show that $H + K$ is a subspace of $V$.
ii. Show that $H$ is a subspace of $H + K$ and $K$ is a subspace of $H + K$.
For part (i) I did check $H+K$ using the zero vector. Here's my work.
$H+K=0$
if $H+K=0$ then $0\in H$ and $0\in K$ which passes the zero vector property because $0+0=0 \in H+K$
Am I on the right track with this? Also, I want to show work for checking the other two checks in subspaces. I don't understand them enough to confidently say I can come up with something but if I had to guess, could I do the following.
Check for closed under vector addition
Let use $H_1 + K_1$ and $H_2 + K_2$.
$H_1 + K_1$ = $H_2 + K_2$
$(H_1 + K_1) + (H_2 + K_2) = 0$ $H_1 + K_1 + H_2 + K_2 = 0$ $H_3 + K_3= 0$Check for closed under vector Multiplication
Let's use c as the scalar.
$c * (H+K)$
$cH+cK$
Can someone explain the last two checks for me (whether I'm right or wrong) please? I don't understand them completely. In what scenario will they prove that a vector is not a subspace type? So far I always get problems where they are all subspaces.
For step (ii) i'm not sure what to do to show that $H$ and $K$ are subspaces of $H+K$