Definition of scalar product in relation to projections

Question

So if $x \in X$ is an element in a vector space $X$, then $\forall x \in X$: $$x = e_ix^i$$ where $e_i$ is a basis for $X$. However, I encountered that this is equivalent to: $$x = e_i (e_i, x)$$ where $(e_i, x)$ is the scalar product between $e_i$ and $x$. Why is the following true? $$x^i = (e_i, x)$$ My lecturer mentioned something about projections, but I don't understand the relation.

Answer 1

Imagine the basis $e_i$ is orthonormal, that is $(e_i, e_j) = \delta_{ij}$ ($1$ if $i = j$ or $0$ otherwise). You can expand any vector $x$ in this bases

$$ x = x^1 e_1 + x^2 e_2 + \cdots = x^ie_i $$

Now multiple both sides by some vector of the basis, say the first one

$$ (e^1, x) = x^1\underbrace{(e_1, e_1)}_{=1} + x^2\underbrace{(e_2, e_1)}_{=0} + x^3\underbrace{(e_3, e_1)}_{=0} + \cdots = x^1 $$

You can repeat the same experiment for all the other components, and quickly realize that

$$ (e^i, x) = x^i $$