Linear and multilinear algebra; matrix theory Definition

Coordinate vector

Builds on Basis

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $B = (b_1, \dots, b_n)$ be a basis of $V$ . Each $v \in V$ is equal to $a_1 b_1 + \cdots + a_n b_n$ for exactly one $(a_1, \dots, a_n) \in F^n$ . The coordinate vector of $v$ in the basis $B$ is $[v]_B = (a_1, \dots, a_n)^T \in F^n.$ The scalars $a_i$ are the coordinates of $v$ in $B$ .

Remarks

The coordinates depend on the order of the basis vectors, so $B$ is an ordered basis. The map $v \mapsto [v]_B$ is a linear isomorphism $V \to F^n$ , see isomorphism with the coordinate space. In $\R^2$ with $B = ((1, 1), (1, -1))$ , the vector $(3, 1)$ has coordinate vector $(2, 1)^T$ . A change of basis matrix converts coordinates in one basis into coordinates in another basis.

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Vector space definition
Linear combination definition
Linear subspace definition
Linear independence definition
Linear span definition
Spanning set definition
Basis definition
Coordinate vector definition

Where this leads

25 pages build on this one.

Tag	`0BMVBHTWV2`
URI	`m2://0BMVBHTWV2`
Kind	definition
Writing	W0 stub
MSC	15A03
Defines	Coordinate vector
Uses	Basis

Coordinate vector

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

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