Let $W \subset V$ with $\dim V= n$. Suppose $w_1,\ldots,w_m$ is an orthogonal basis for $W$ and $w_{m+1},\ldots,w_n$ is an orthogonal basis for $W^\perp$.
a.) Prove that the combination $w_1,\ldots,w_n$ form an orthogonal basis of $V$.
b.) Show that if $v=c_1w_1+\cdots+c_nw_n$ is any vector in $V$, then its orthogonal decomposition $v=w+z$ is given by $w=c_1w_1 + \cdots+c_mw_m \in W$ and $z=c_{m+1}w_{m+1}+\cdots+c_nw_n\in W^\perp$
How will I be able to prove this?
I know that if $\dim W=m$ and $\dim V=n$, then $\dim W^\perp = n-m$ and since $W\subset V$ then its orthogonal basis $w = w_1,\ldots,w_m$ is an orthogonal complement of $V$ iff $\langle w_i,v_i \rangle = 0$, but how will I be able to prove that using the conditions given in the question?