I am wondering if someone can check my proof that the sum of two subspaces is a subspace:
1) First show that $0 \in W_1 + W_2$:
Since $W_1, W_2$ are subspaces, we know that $0 \in W_1, W_2$.
So if $w_1, w_2 = 0, w_1 + w_2 = 0 + 0 = 0 \in W_1 + W_2$.
2) Show that $cu+v \in W_1 + W_2$.
Let $W = W_1 + W_2$
Let $cu \in W$. We can find some $u_1 \in W_1, u_2 \in W_2, c \in \mathbb{F}$ such that $cu = u_1 + u_2$.
Similarly, since $v \in W$, we can find $v_1, v_2 \in W_1, W_2$ such that $v = v_1 + v_2$.
So $cu+v = c(u_1 + u_2) + (v_1 + v_2)$
$= cu_1 + cu_2 + v_1 + v_2$
$= cu_1 + v_1 + cu_2 + v_2$
And so $cu_1 + v_1 \in W_1$ and $cu_2 + v_2 \in W_2$, and therefore $cu+v \in W_1 + W_2$.
Is my proof correct? Please let me know!
Best Answer
Your proof is not rigorously correct.
The first part 1) is perfect.
For the part 2), there are some points that can be improved.
You should focus in these details in order to obtain a logically correct proof.