I have a couple Linear Algebra proofs that I'm having trouble getting through. I was wondering if someone could help walk me through them. They're harder than most of what we've done so far.
1) Suppose \(\displaystyle V\) is an inner product space with dimension \(\displaystyle n\). Prove that any set of \(\displaystyle n\) nonzero orthogonal vectors form a basis for \(\displaystyle V\) .
2) Let \(\displaystyle B= [ \vec u_1, \vec u_2, ..., \vec u_p ]\) be an orthonormal basis for a subspace \(\displaystyle W\). Let \(\displaystyle \vec v\) be any vector in \(\displaystyle W\), where \(\displaystyle \vec v=k_1\vec u_1 + k_2\vec u_2 + ... + k_p\vec u_p\). Prove that \(\displaystyle ||v||^2=k_1^2+k_2^2+...+k_p^2\).
1) Suppose \(\displaystyle V\) is an inner product space with dimension \(\displaystyle n\). Prove that any set of \(\displaystyle n\) nonzero orthogonal vectors form a basis for \(\displaystyle V\) .
2) Let \(\displaystyle B= [ \vec u_1, \vec u_2, ..., \vec u_p ]\) be an orthonormal basis for a subspace \(\displaystyle W\). Let \(\displaystyle \vec v\) be any vector in \(\displaystyle W\), where \(\displaystyle \vec v=k_1\vec u_1 + k_2\vec u_2 + ... + k_p\vec u_p\). Prove that \(\displaystyle ||v||^2=k_1^2+k_2^2+...+k_p^2\).