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?
Best Answer
For the first part, let us call the first basis $\mathcal{B}$ and the second $\mathcal{C}$. Then we're interested in showing $\mathcal{B}\cup\mathcal{C}$ is an orthogonal basis. $\mathcal{B}$ itself is orthogonal and so is $\mathcal{C}$. Now because these are basis of orthogonal complements you can show that $w_i \perp w_j$ for $i\le m$ and $j > m$. This will show that the vectors are pairwise orthogonal. Orthogonal (non-zero) vectors always form a linearly independent set.
For the second part, first show that the given decomposition is a valid decomposition. Then suppose that $\mathbf{v}$ can be decomposed in two ways $$\mathbf{v} = \mathbf{w}+\mathbf{z} = \mathbf{w'}+\mathbf{z'}$$ with the $\bf{w}$s in $W$ and the $\bf{z}$s in $W^\perp$. Then what can you say about $$\mathbf{w}-\mathbf{w'} = \mathbf{z}' - \mathbf{z}$$
I suggest that you follow the above and fill in all the details with care. I've skipped over statements which may require proof.