Let $V = U\oplus W = U_{1}\oplus W_{1}$ and $U_{1}\subseteq U$ and $W_{1}\subseteq W$. Show that $U_{1} = U$ and $W_{1} = W$.

Question

Let $V = U\oplus W = U_{1}\oplus W_{1}$ and $U_{1}\subseteq U$ and $W_{1}\subseteq W$. Show that $U_{1} = U$ and $W_{1} = W$.

Answer 1

If $v \in U \oplus W$, then there exists $u \in U$ and $w \in W$ such that $v = u + w \in U \oplus W.$

Since $U \oplus W = U_1 \oplus W_1$, there exists $u_1 \in U_1$ and $w_1 \in W_1$ such that

$$u + w = u_1 + w_1 \iff u - u_1 = w - w_1$$

Now $u - u_1 \in U$ and $w - w_1 \in W$ andd since the sum is direct, we have $U \cap W = \{ 0 \}$. Hence $u = u_1$ and $w = w_1.$ The elements $u$ and $w$ were chosen arbitrary, the result now follows.

Answer 2

if $V$ is of finite dimension, another possible solution is to say that

$\dim V=\dim U +\dim W=\dim U_{1} +\dim W_{1}$

and

$\dim U_{1} \le \dim U, \dim W_{1} \le \dim W$ with both equal iff $U=U_{1}$ and $W=W_{1}$

but they must be equal for the assumption to hold, and that is it.