# Notes - Linear Algebra MT23, Orthogonal sets

### Flashcards

Quickly prove that any orthonormal set $\{u _ 1, \cdots, u _ n \}$ is linearly independent.

Suppose

\[\lambda_1 u_1 + \cdots + \lambda_m u_m = 0\]Take $\langle u _ j, \cdot\rangle$ for each $u _ j$, and you see

\[\lambda_j = 0\]Suppose $\{u _ 1, \cdots, u _ n\}$ is an orthonormal basis, so that every $v$ can be written

\[v = \lambda_1 u_1 + \cdots + \lambda_n u_n\]
How can you write $\langle u _ j, v \rangle$?

\[\langle u_j, v \rangle = \lambda_j\]

Suppose $\{u _ 1, \cdots, u _ n\}$ is an orthonormal basis, so that every $v$ can be written

\[v = \lambda_1 u_1 + \cdots + \lambda_n u_n\]
Can you write $v$ as a sum of inner products with elements of the basis?

\[\sum^n_{j=1} \langle u_j, v \rangle u_j\]