Chapter 23. Coordinate Systems

A coordinate system is a way to describe vectors by scalars. In linear algebra, coordinates are not attached to a vector by nature. They depend on a chosen basis.

A vector is an element of a vector space. A coordinate vector is a column of scalars that represents that vector relative to an ordered basis. The same vector can have different coordinate vectors under different bases. A basis gives each vector a unique coordinate representation.

23.1 Coordinates from a Basis

Let $V$ be a vector space over a field $F$. Let

be an ordered basis of $V$.

Since $B$ spans $V$, every vector $v\in V$ can be written as

Since $B$ is linearly independent, the scalars

are unique.

The coordinate vector of $v$ with respect to $B$ is

The notation $[v]_B$ means “the coordinates of $v$ in the basis $B$.”

23.2 Ordered Bases

A coordinate system requires an ordered basis.

The set

and the set

contain the same vectors, but they give different coordinate columns.

If

then with

we have

With

the same vector is written as

so

The order of the basis determines the order of the coordinates.

23.3 Standard Coordinates in $\mathbb{R}^n$

The standard basis of $\mathbb{R}^n$ is

where

Every vector

has the expansion

Therefore

In the standard basis, the coordinate vector looks exactly like the vector. This is why it is easy to forget that coordinates depend on a basis.

23.4 A Nonstandard Coordinate System

Consider the basis

of $\mathbb{R}^2$.

Let

To find $[x]_B$, solve

This gives

Thus

Adding the equations gives

so

Then

Therefore

The vector has standard coordinates $(7,3)$, but its coordinates in $B$ are $(5,2)$.

23.5 Coordinate Maps

Given an ordered basis

of $V$, the coordinate map is the function

defined by

This map sends an abstract vector to its coordinate column.

It is linear. If

then

Therefore

Similarly,

Thus coordinates preserve the vector space operations.

23.6 Coordinates and Isomorphism

The coordinate map

is an isomorphism.

It is one-to-one because each vector has unique coordinates. It is onto because every column

comes from the vector

Thus every $n$-dimensional vector space over $F$ can be identified with $F^n$ after choosing a basis.

This identification is useful, but it should be read carefully. The space $V$ and the coordinate space $F^n$ are structurally equivalent, but the equivalence depends on the chosen basis.

23.7 Coordinate Matrix of a Basis

Let

be a basis of $\mathbb{R}^n$. Place the basis vectors as columns of a matrix:

Then for any vector $x\in\mathbb{R}^n$,

This equation converts $B$-coordinates into standard coordinates.

The matrix $P_B$ is invertible because its columns form a basis. Therefore

This equation converts standard coordinates into $B$-coordinates.

23.8 Example Using a Basis Matrix

Let

Then

For

standard coordinates are

Conversely, if

then

Since

we get

23.9 Change of Coordinates

A change of coordinates converts the coordinate vector of the same vector from one basis to another.

Let

and

be bases of $V$.

For a vector $v$, one may know

and want

The vector $v$ itself does not change. Only its coordinate description changes.

In $\mathbb{R}^n$, if $P_B$ and $P_C$ are the basis matrices, then

and

Thus

Multiplying by $P_C^{-1}$,

The matrix

is called the change-of-coordinates matrix from $B$-coordinates to $C$-coordinates.

23.10 Identity Map, Different Coordinates

The formula

represents the identity map on $V$, not a transformation of the vector itself.

The vector remains $v$. The coordinate column changes because the basis changes.

This distinction is important. A matrix can represent either:

The same numerical matrix may appear in either role, but the interpretation differs.

23.11 Coordinates in Polynomial Spaces

Let

be the vector space of polynomials of degree at most $2$. Let

For

the coordinate vector is

Now choose another basis:

To find $[p]_C$, solve

The right-hand side is

Match coefficients with

Then

so

and

so

Thus

The polynomial has not changed. Its coordinates changed because the basis changed.

23.12 Coordinates in Matrix Spaces

Let

have the ordered basis

where

For

we have

Therefore

A matrix can itself be a vector. Its coordinates are determined by the ordered basis chosen for the matrix space.

23.13 Coordinates in Function Spaces

Let

with ordered basis

For

the coordinate vector is

Again, coordinates record coefficients relative to a basis. The vector may be a function, a polynomial, a matrix, or another object. The coordinate principle is the same.

23.14 Coordinate Systems and Linear Equations

Coordinate systems convert vector equations into scalar equations.

Suppose

Finding the coordinates of $v$ means solving for the unknown scalars $c_1,\ldots,c_n$.

In $\mathbb{R}^n$, this becomes the linear system

where

Because $P_B$ is invertible, the system has a unique solution.

Thus coordinate computation is a special case of solving a linear system.

23.15 Coordinate Systems and Geometry

In geometry, a coordinate system assigns numbers to points or vectors. In linear algebra, the origin and basis vectors determine the coordinate system for a vector space.

For vector spaces, the origin is fixed as the zero vector. A basis determines the coordinate axes.

In $\mathbb{R}^2$, the standard basis gives the usual horizontal and vertical axes. A nonstandard basis gives oblique axes. Coordinates then measure how far to move along each basis direction.

If the basis vectors are perpendicular and have length $1$, the coordinate system is orthonormal. If they are not perpendicular, the coordinate system is oblique.

Both are valid linear coordinate systems as long as the basis vectors are linearly independent and span the space.

23.16 Coordinate Systems and Computation

Coordinates are essential for computation because computers manipulate arrays of numbers.

An abstract vector space becomes computationally usable after choosing a basis. Polynomials become coefficient arrays. Matrices become flattened arrays. Functions in finite-dimensional subspaces become coefficient vectors.

For example, the polynomial

can be stored as

This coordinate representation allows addition, scalar multiplication, differentiation, integration, and evaluation to be implemented as operations on arrays.

23.17 Coordinate Dependence

Some quantities depend on the chosen coordinate system. Others do not.

The coordinate vector $[v]_B$ depends on $B$. The vector $v$ itself does not.

The matrix of a linear transformation depends on the chosen bases. The transformation itself does not.

Dimension does not depend on the basis. Linear independence of an actual list of vectors does not depend on coordinates, although coordinates may be used to test it.

This distinction leads to an important habit: separate objects from their representations.

23.18 Standard Coordinates Are Not Intrinsic

In $\mathbb{R}^n$, the standard coordinates are so familiar that they may seem intrinsic. They are not. They are coordinates relative to the standard basis.

The vector

really means

with respect to the standard basis.

Under a different basis, the same vector may have coordinates

or another column. The column alone does not fully describe the vector unless the basis is known.

A coordinate vector always has an implicit basis.

23.19 Common Notation

23.20 Summary

A coordinate system is determined by an ordered basis. It represents each vector by the unique scalars needed to express that vector as a linear combination of the basis vectors.

The key ideas are:

Coordinates make vector spaces computable. They turn abstract vectors into columns of scalars. But coordinates are representations, not the underlying objects. The basis is part of the meaning.