Exercise 3.4.43

The problem is to show that the $u$ ’s, $v$ ’s, $w$ ’s together are independent. We know the $u$ ’s and $v$ ’s together are a basis for $V$, and the $u$ ’s and $w$ ’s together are a basis for $W$. Suppose a combination of $u$ ’s, $v$ ’s, $w$ ’s gives $0$. To be proved: All coefficients $=$ zero.

Key idea: In that combination giving $0$, the part $x$ from the $u$ ’s and $v$ ’s is in $V$. So the part from the $w$ ’s is $-x$. This part is now in $V$ and also in $W$. But if $-x$ is in $V\cap W$ it is a combination of $u$ ’s only. Now the combination giving $0$ uses only $u$ ’s and $v$ ’s (independent in $V!$ ) so all coefficients of $u$ ’s and $v$ ’s must be zero. Then $x=0$ and the coefficients of the $w$ ’s are also zero.