What is $W_1 + W_2$ if $W_1\subset W_2$?

Question

If $W_1$ and $W_2$ are vector subspaces. What is $W_1 + W_2$ if $W_1\subset W_2$ ?

I have assumed that $v\in W_1 + W_2$, and proved that $(W_1 + W_2) \subset W_2$? Is this the correct answer or am I only half way through the solution?

Answer 1

Hint: What is the definition of the direct sum $U+V$?

Answer 2

The set of all sums $w_1+w_2$ are all in $W_2$. The reverse containment is also trivial. So the answer is $W_2$.

Answer 3

If $W_1 \subset W_2$, then we actually cannot write $W_1 \oplus W_2$, because this does not fit the definition of $\oplus$. If we talk about $W_1 + W_2$, then by definition this is just $W_2$ by definition of $+$ between vector subspaces.