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Core Concept Map Language

Theorems from Coordinates and bases

All theorems from Coordinates and bases.

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[THEOREM]

Orthonormal sets are independent

Every orthonormal set is linearly independent.

If $\lambda_1u_1+\cdots+\lambda_nu_n=0$ , take the dot product with $u_k$ . Orthonormality leaves $\lambda_k=0$ . This holds for every $k$ .

[1]:Orthonormal set
[2]:Linear independence

[THEOREM]

Coordinates in an orthonormal basis

If $B=(u_1,\ldots,u_n)$ is an ordered orthonormal basis of a subspace $V\subseteq\mathbb R^m$ and $v\in V$ , then

[v]_B= \begin{pmatrix} u_1\cdot v\\ \vdots\\ u_n\cdot v \end{pmatrix}.

Each dot product extracts one coefficient because the other basis vectors are perpendicular to it.

[1]:Orthonormal basis
[2]:Coordinate vector
[3]:Proof
[4]:An orthonormal basis of $\mathbb R^3$

[THEOREM]

Coordinates preserve vector-space operations

If $B$ is an ordered basis for $V$ , $u,v\in V$ and $\lambda\in F$ , then

u=v\iff [u]_B=[v]_B, \qquad [u+v]_B=[u]_B+[v]_B, \qquad [\lambda u]_B=\lambda[u]_B.

These identities follow by adding or scaling the coefficients in the unique representations.

[1]:Coordinate vector
[2]:Basis

This page builds on Basis and dimension, Linear independence and Spans.

It leads to Matrix representations (Linear transformations), Data fitting and polynomial interpolation and Seen Questions (Linear transformations).

Part of the thread Coordinates and linear structure, in 5 places.

A basis turns vectors and maps into columns and matrices; a good basis makes the calculation simple.

DFCoordinate vectorcoordinates in a basisALG 2 / Here

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Orthonormal sets are independent Coordinates in an orthonormal basis Coordinates preserve vector-space operations