# Basis linear algebra question

• Nov 27th 2013, 06:37 PM
MichaelH
Basis linear algebra question
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 $v_1,\cdots,v_n \in \mathbb{R}^n$ is an orthonormal basis if the following holds for all $i, j = 1,\cdots, n$:

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

Assuming this, I have to:

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

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

b) Show that:

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

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

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

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

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

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

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

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

As I said, this seems like a lot to go through and appreciate any help since I literally am stuck!
• Nov 27th 2013, 06:38 PM
topsquark
Re: Basis linear algebra question

-Dan
• Nov 27th 2013, 06:39 PM
MichaelH
Re: Basis linear algebra question
Quote:

Originally Posted by topsquark