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**Runty** Let U and W be orthogonal subspaces.

a) Prove that $\displaystyle U\cap W=\{\underline{0} \}$

If $\displaystyle x\in U\cap W$ then as $\displaystyle x\in U\,\,\,and\,\,\,x\in W$ we get $\displaystyle <x,x>=0\Longleftrightarrow x=0$ , with <,> the inner product in the v.s.

b) If $\displaystyle \{u_1,u_2,...,u_m\}$ is a linearly independent subset of U and $\displaystyle \{w_1,w_2,...,w_n\}$ is a linearly independent subset of W, prove that $\displaystyle \{u_1,u_2,...,u_m,w_1,w_2,...,w_n\}$ is a linearly independent set in V.

Suppose $\displaystyle \sum^m_{i=1}a_1u_1+\sum^n_{j=1}b_jw_j=0\,,\,\,a_i\ ,,\,b_j \in\mathbb{F}=$ the definition field, then $\displaystyle \sum^m_{i=1}a_1u_1=-\sum^n_{j=1}b_jw_j\Longrightarrow \sum^m_{i=1}a_1u_1\,,\,\sum^n_{j=1}b_jw_j \in U\cap W$ $\displaystyle \Longrightarrow \sum^m_{i=1}a_1u_1=0=\sum^n_{j=1}b_jw_j\Longrighta rrow$ . Deduce from here and (a) above that $\displaystyle a_i=0=b_j\,\,\forall\,i,j$ .

Tonio

I find this bit a little hard to understand, and I have to multitask, so I'll have difficulty if I have to do this on my own. Any help would be greatly appreciated.