I have a really long homework question that I need help with. There are a couple of parts to it so I appreciate anyone that can help out.

It starts by just stating that a basis $\displaystyle v_1,\cdots,v_n \in \mathbb{R}^n$ is an orthonormal basis if the following holds for all $\displaystyle i, j = 1,\cdots, n$:

$\displaystyle v_i \cdot v_j = \begin{cases} 0 & i \neq j \\ 1 & i=j \end{cases}$

Assuming this, I have to:

a) Show that for any $\displaystyle x \in \mathbb{R}^n$ we have:

$\displaystyle x=(v_1 \cdot x)v_1+ \cdots +(v_n \cdot x)v_n$

b) Show that:

$\displaystyle v_1 = \frac{1}{sqrt{2}} (1, 0, 1)$

$\displaystyle v_2 = \frac{1}{sqrt{3}} (1, 1, -1)$

$\displaystyle v_3 = \frac{1}{sqrt{6}} (-1, 2, 1)$

is an orthonormal basis in [tex]\mathbb{R}^3[\tex] and expand the vector $\displaystyle x=(10, -12, 3)$ in the basis $\displaystyle v_1, v_2, v_3$.

c) Show that for any $\displaystyle x \in \mathbb{R}^n$:

$\displaystyle x \cdot x = (v_1 \cdot x)^2 + \cdots + (v_n \cdot x)^2$

and for any $\displaystyle x, y \in \mathbb{R}^n$:

$\displaystyle x \cdot y = (v_1 \cdot x)(v_1 \cdot y) + \cdots + (v_n \cdot x)(v_n \cdot y)$

As I said, this seems like a lot to go through and appreciate any help since I literally am stuck!