Why is orthonormal basis better than linearly independent basis and orthogonal basis?

Question

My course teacher for Pattern Recognition course made this statement that orthonormal basis is better as compared to orthogonal or linearly independent basis when it comes to pattern recognition, or linear algebra in general, but she didn't tell us why!

I know that orthonormal basis consists of unit vectors which are orthogonal to each other. But what makes them 'better'? Or in which scenario do the above mentioned other two basis fail and orthonormal basis survive?

Do orthonormal basis make analyzing things easier? How? Any help is appreciable. Thanks in advance.

Answer 1

Suppose you seek to express a vector $a$ as a linear combination of members of an orthonormal basis $e_1,e_2,e_3,\ldots$ It is as follows: $$ (a\cdot e_1) e_1 + (a\cdot e_2) e_2 + (a\cdot e_3) e_3 + \cdots. $$ Just try doing it with a basis that is not orthogonal. You'll see that it's far more complicated. And it's also somewhat more complicated with a basis that is orthogonal but not orthonormal.

Answer 2

If the coordinates of a vector $v$ with respect to an orthonormal basis are $\alpha_1,\alpha_2,\ldots,\alpha_n$, then$$\|v\|=\sqrt{{\alpha_1}^2+{\alpha_2}^2+\cdots+{\alpha_n}^n}.$$And if $\beta_1,\beta_2,\ldots,\beta_n$ are the coordinates of another vector, $w$, then$$\langle v,w\rangle=\sum_{k=1}^n\alpha_k\beta_k.$$