Orthogonal Sets

If for set $\{{\vec{u}}_1,\dots ,{\vec{v}}_n\}$ for $j\ne k$, ${\vec{u}}_j\perp {\vec{u}}_k$. i.e. ${\vec{u}}_j\cdot {\vec{u}}_k=0$

If Set S is orthogonal, the vectors of S are linearly independent.

Expansion in Orthogonal Basis

If we have an Orthogonal Basis $\{{\vec{u}}_1,\dots ,{\vec{v}}_n\}$ in ${\mathbb{R}}^n$ then for any $\vec{w}\in {\mathbb{R}}^n$,

$$\vec{w}=c_1{\vec{u}}_1+\cdots +c_n{\vec{v}}_n$$

$C_q$ can be found using $c_q=\frac{\vec{w}\cdot {\vec{u}}_q}{{\vec{u}}_q\cdot {\vec{u}}_q}$

Length of a Vector in the basis of an orthogonal set

If $\vec{w}=c_1{\vec{u}}_1+\cdots +c_n{\vec{v}}_n=(\vec{w}\cdot {\vec{u}}_1){\vec{u}}_1+\cdots +(\vec{w}\cdot {\vec{u}}_n){\vec{v}}_n$

$$||\vec{w}||=\sqrt{(\vec{w}\cdot {\vec{u}}_1{)}^2+\cdots +(\vec{w}\cdot {\vec{u}}_n{)}^2}$$