Proof: Sum of dimension of orthogonal complement and vector subspace

Let $\beta=\{w_1,w_2,\ldots,w_k\}$ and $\gamma=\{x_1,x_2,\ldots,x_m\}$ be the bases for $W$ and $W^\perp$, respectively. It suffices to show that $$\beta\cup\gamma=\{w_1,w_2,\ldots,w_k,x_1,x_2,\ldots,x_m\}$$ is a basis for $V$. Given $v\in V$, then it is well-known that $v=v_1+v_2$ for some $v_1\in W$ and $v_2\in W^\perp$. Also because $\beta$ and $\gamma$ are bases for $W$ and $W^\perp$, respectively, there exist scalars $a_1,a_2,\ldots,a_k,b_1,b_2,\ldots,b_m$ such that $v_1=\displaystyle\sum_{i=1}^ka_iw_i$ and $v_2=\displaystyle\sum_{j=1}^mb_jx_j$. Therefore $$v=v_1+v_2=\sum_{i=1}^ka_iw_i+\sum_{j=1}^mb_jx_j,$$ which follows that $\beta\cup\gamma$ generates $V$. Next, we show that $\beta\cup\gamma$ is linearly independent. Given $c_1,c_2,\ldots,c_k,d_1,d_2,\ldots,d_m$ such that $\displaystyle\sum_{i=1}^kc_iw_i+\sum_{j=1}^md_jx_j={\it 0}$, then $\displaystyle\sum_{i=1}^kc_iw_i=-\sum_{j=1}^md_jx_j$. It follows that $$\sum_{i=1}^kc_iw_i\in W\cap W^\perp\quad\mbox{and}\quad \sum_{j=1}^md_jx_j\in W\cap W^\perp.$$ But since $W\cap W^\perp=\{{\it 0}\,\}$ (gievn $x\in W\cap W^\perp$, we have $\langle x,x\rangle=0$ and thus $x={\it 0}\,$), we have $\displaystyle\sum_{i=1}^kc_iw_i=\sum_{j=1}^md_jx_j={\it 0}$. Therefore $c_i=0$ and $d_j=0$ for each $i,j$ becasue $\beta$ and $\gamma$ are bases for $W$ and $W^\perp$, respectively. Hence we conclude that $\beta\cup\gamma$ is linearly independent.

Hint:

Take a basis $w_1,\dots,w_r$ of $W$, and consider the linear forms on $V$ defined by $w_i^*:v\mapsto\langle w_i,v\rangle$.

These linear forms are linearly independent, hence the solutions of the system of equations $w_i^*(v)=0,\ i=1,\dots r$ has codimension $r$ by the rank-nullity theorem. These solutions are precisely the orthogonal complement $\;U^{\bot}$.

It is sufficient to show that $V=W\oplus W^{\perp}$. If $v\in W\cap W^{\perp}$, then $\left\langle v,v\right\rangle=0$. Hence it remains to show that any vector $v\in V$ can be written as $v=w+w'$ with $w\in W$ and $w'\in W^{\perp}$.

Choose an orthonormal basis $\left\{w_1,\dots , w_k\right\}$ of $W$ and extend to an orthonormal basis $\left\{w_1,\dots,w_k,v_{k+1},\dots ,v_n\right\}$ of $V$. By definition $v_i\in W^{\perp}$ for all $n\geq i\geq k+1$. Hence any $v\in V$ can be decomposed as we needed to show.