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?


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

Proofs




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