I have an orthogonal subset of nonzero vectors $u_1,\dots,u_n$. I take $v \in V$ (a vector space) so that $v$ is the in the span of the previous set. Now I let $u$ be $v-m_1u_1-\dots-m_nu_n$ with $m_i$ real numbers.
I need to calculate $\langle u,u_i\rangle$ (an inner product) for $i=1,\dots,n$ and then explain how to choose $m_i$ so that this inner product is $1$.
Then I need to deduce from the result that if $v_1,\dots,v_n$ is a basis of $V$ then there is an orthogonal subset of $V$, $u_1,\dots,u_n$, such that $\operatorname{span}[v_1,\dots,v_i]=\operatorname{span}[u_1,\dots,u_i]$ for all $i$ in $[1,n]$.
I think I've done the first one right but I can't find out the deduction part of the problem.
Any help would be much appreciated! Thanks
For the first part, you have $$ \langle u,u_i\rangle= \langle v,u_i\rangle -m_1\langle u_1,u_i\rangle -m_2\langle u_n,u_i\rangle -\dots -m_n\langle u_n,u_i\rangle = \langle v,u_i\rangle -m_i\langle u_i,u_i\rangle =1 $$ so it should be easy to find out what $m_i$ is. But shouldn't it be $0$?
Part two is the usual Gram-Schmidt construction. Starting from $u_1=v_1$ you compute $u_2$ as $v_2$ minus its orthogonal projection on $\operatorname{span}[u_1]$; in the generic step, after having found $u_1,\dots,u_i$ (with $i<n$) you choose $u_{i+1}$ as $v_{i+1}$ minus its orthogonal projection on $\operatorname{span}[u_1,\dots,u_n]$.
This is exactly like solving for $m_1, m_2,\dots,m_i$ as before (but with final result $0$ and not $1$).