I was looking at a proof of Gram Schmidt theorem and I saw the following lemma, it starts here:
First the theorem: if $V$ is an inner product space and $X= \{x_1,\dots, x_n\}$ is a linearly independent set then there exist an orthonormal set $Y=\{y_1,\dots, y_n\}$ such that $\operatorname{span}(X)=\operatorname{span}(Y)$.
The lemma used goes like this:
Let $Z = \{z_1, \dots, z_n\}$ and be an orthonormal set and $v \notin \operatorname{span}(Z)$. Then $$w = v- \sum^n_{i=1}\langle v, z_i\rangle z_i$$ is orthogonal to $z$.
No problem so far. It's the following calculation I wanted to be sre I wasn't missing a stupid obvious thing: $\langle w, z_j \rangle = \langle (v- \sum^n_{i=1}\langle v, z_i\rangle z_i, z_j\rangle=\langle v, z_j \rangle - \sum^n_{i=1}\langle v, z_i\rangle \langle z_i, z_j\rangle = \langle v, z_j \rangle - \sum^n_{j=1}\delta_{ij} \langle v, z_j\rangle = 0$
It's the very last step I wondered about. Why is it the Kroeneker delta? Is is some property of the inner product multiplication? (I know that odds are there is an algebraic step the text just skipped but I just want to be dead sure I know how they got it). My guess was that if you multiply a vector like that by itself you get 1 but I wasn't sure.
thanks in advance. I know it's sort of dumb, but I don't like it when I feel I have to take someone's word for stuff I don't quite get.