Exercise 1.3.30

Proof.

$⟹$

If $V=W_1\oplus W_2$ and some vector $y\in V$ can be represented as $y=x_1+x_2=x_1^{\prime }+x_2^{\prime }$, where $x_1,x_1^{\prime }\in W_1$ and $x_2,x_2^{\prime }\in W_2$, then we have $x_1-x_1^{\prime }\in W_1$ and $x_1-x_1^{\prime }=x_2+x_2^{\prime }\in W_2$. But since $W_1\cap W_2=\{0\}$, we have $x_1=x_1^{\prime }$ and $x_2=x_2^{\prime }$.

$⟸$

Conversely, if each vector in $V$ can be uniquely written as $x_1+x_2$, then $V=W_1+W_2$. Now if $x\in W_1\cap W_2$ and $x\ne 0$, then we have that $x=x+0$ with $x\in W_1$ and $0\in W_2$ or $x=0+x$ with $0\in W_1$ and $x\in W_2$, a contradiction.