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Core Concept Map Language

Definitions from Coordinates and bases

All definitions from Coordinates and bases.

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[DEFINITION]

Coordinate vector

Let $B=(u_1,\ldots,u_n)$ be an ordered basis for a vector space $V$ . If

u=x_1u_1+\cdots+x_nu_n,

then the coordinate vector of $u$ with respect to $B$ is the column

[u]_B= \begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix}.

The order of $B$ fixes the order of the entries. Changing the order of the basis changes the coordinate vector even though it does not change $u$ .

[1]:Basis
[2]:Independence and uniqueness
[3]:Matrix representation of a linear map
[4]:Construct a matrix representation
[5]:Matrix representations
[6]:Compute a coordinate vector
5 more

[DEFINITION]

Orthonormal set

A set $S=\{u_1,\ldots,u_n\}$ in $\mathbb R^m$ is orthonormal if

u_i\cdot u_j= \begin{cases} 1,&i=j,\\ 0,&i\ne j. \end{cases}

[1]:Orthonormal sets are independent
[2]:Orthonormal basis

[DEFINITION]

Orthonormal basis

An orthonormal basis of a subspace of $\mathbb R^m$ is a basis that is also an orthonormal set.

[1]:Basis
[2]:Orthonormal set
[3]:Real symmetric matrices
[4]:Coordinates in an orthonormal 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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Coordinate vector Orthonormal set Orthonormal basis