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?
\[\sum^n _ {j=1} \langle u _ j, v \rangle u _ j\]