Coordinate Systems

Choosing a basis for a vector space gives it a coordinate system. Each vector acquires a unique list of weights, and those weights behave exactly like a column in ${\mathbb{R}}^n$. This is why every finite-dimensional space, however abstract, can be computed with as if it were ${\mathbb{R}}^n$: the correspondence preserves all vector-space structure.¹

Unique representation

The uniqueness of coordinates is what makes them a coordinate system rather than just one of many ways to write a vector.

Existence comes from spanning; uniqueness from independence. If $x$ also equalled $d_1{b}_1+\cdots +d_n{b}_n$, subtracting gives $0=(c_1-d_1)b_1+\cdots +(c_n-d_n)b_n$, and independence forces every $c_j-d_j=0$. So the weights are determined by $x$ and $\mathcal{B}$ alone.

The basis must be an ordered set, so that the entries of $[x{]}_{\mathcal{B}}$ have a fixed meaning. For the standard basis $\mathcal{E}$ of ${\mathbb{R}}^n$, the coordinate vector is the vector itself: $[x{]}_{\mathcal{E}}=x$.

Coordinates as a change of grid

Coordinates are the readout of a grid drawn from the basis vectors. Standard graph paper is the grid of the standard basis; a different basis rules a different grid over the same plane, and the same point gets different coordinates on each.

The change-of-coordinates matrix in ${\mathbb{R}}^n$

When $\mathcal{B}=\{b_1,\dots ,b_n\}$ is a basis for ${\mathbb{R}}^n$, collecting the basis vectors as columns gives a matrix that converts $\mathcal{B}$-coordinates back to standard coordinates. Let

Then $x=c_1{b}_1+\cdots +c_n{b}_n$ rewrites as the matrix statement

$P_{\mathcal{B}}$ is the change-of-coordinates matrix from $\mathcal{B}$ to the standard basis. Its columns are a basis, so it is invertible, and

Left-multiplication by $P_{\mathcal{B}}$ goes from basis coordinates to standard coordinates; by $P_{\mathcal{B}}^{-1}$ goes the other way.

Converting between two nonstandard bases is the subject of change of basis.

The coordinate mapping is an isomorphism

The coordinate mapping respects every vector-space operation.

Linearity is a direct computation. If $u=c_1{b}_1+\cdots +c_n{b}_n$ and $w=d_1{b}_1+\cdots +d_n{b}_n$, then $u+w=(c_1+d_1)b_1+\cdots +(c_n+d_n)b_n$, so

The mapping preserves addition and scalar multiplication, and by uniqueness it is one-to-one and onto. More generally the coordinate vector of a linear combination is the same combination of the coordinate vectors:

The coordinate-mapping theorem says the coordinate mapping is an isomorphism from $V$ onto ${\mathbb{R}}^n$. Any real vector space with a basis of $n$ vectors is isomorphic to ${\mathbb{R}}^n$, so the abstract space and the concrete one carry identical vector-space structure under different names.

Polynomials become columns

Let $\mathcal{B}=\{1,t,t^2,t^3\}$ be the standard basis of ${\mathbb{P}}_3$. A polynomial $p(t)=a_0+a_{1}t+a_2{t}^2+a_3{t}^3$ is already written in the basis, so $[p{]}_{\mathcal{B}}=(a_0,a_1,a_2,a_3)$, and $p\mapsto [p{]}_{\mathcal{B}}$ is an isomorphism ${\mathbb{P}}_3\to {\mathbb{R}}^4$. Every polynomial operation corresponds to a column operation.

Questions about polynomials become questions about columns.

A plane isomorphic to ${\mathbb{R}}^2$

Coordinates also work inside a subspace.

Whether a different basis for $H$ produces the same conclusion — that $H$ is isomorphic to ${\mathbb{R}}^2$ rather than some other ${\mathbb{R}}^k$ — is settled by dimension, which shows the number of basis vectors does not depend on the basis chosen.