Coordinates and Change of Basis

Modern interactive lesson

Coordinates and Change of Basis

Translate vectors between coordinate systems. Compute basis coordinates, change-of-basis matrices, and the representation of a linear map in a new basis.

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Big idea: vector versus address

The object

A vector is the geometric object itself. In the standard basis of $\mathbb{R}^2$, $x=\begin{bmatrix}3\\1\end{bmatrix}$ means $3e_1+e_2$.

The address

Coordinates depend on the basis. If $\mathcal B=\{(1,1),(1,-1)\}$, then the same vector has $[x]_{\mathcal B}=\begin{bmatrix}2\\1\end{bmatrix}$.

Chapter slogan: The vector does not change. The coordinate system changes.

Basis matrix

For a basis $\mathcal B=\{v_1,\dots,v_n\}$, put the basis vectors as columns:

$$P_{\mathcal B}=\begin{bmatrix}|&&|\\v_1&\cdots&v_n\\|&&|\end{bmatrix}.$$

Basis coordinates to standard coordinates

$$x=P_{\mathcal B}[x]_{\mathcal B}.$$

Standard coordinates to basis coordinates

$$[x]_{\mathcal B}=P_{\mathcal B}^{-1}x.$$

Why columns, not rows?

The expression $x=c_1v_1+\cdots+c_nv_n$ is exactly the matrix product $P_{\mathcal B}c$ when the basis vectors are columns.

Interactive coordinate lab

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Choose a basis $\mathcal B=\{v_1,v_2\}$ and a vector $x$. The page computes $[x]_{\mathcal B}$ and draws the coordinate grid.

$v_1=(a,b)$

$v_2=(c,d)$

$x=(x_1,x_2)$

build from coordinates $(c_1,c_2)$

Blue and purple arrows are basis vectors. Green is the target vector. The tilted grid is the coordinate grid generated by the basis.

Change-of-basis machine

$B\to C$

Start with coordinates in basis $\mathcal B$, convert to the actual vector $x$, then translate that vector into basis $\mathcal C$.

Basis $\mathcal B$: rows are $v_1;v_2$

Basis $\mathcal C$: rows are $w_1;w_2$

$[x]_{\mathcal B}$

Matrix representation in a new basis

If $A$ is the standard matrix of a transformation and $P_{\mathcal B}$ is the basis matrix, then the matrix that acts on $\mathcal B$-coordinates is

$$[A]_{\mathcal B}=P_{\mathcal B}^{-1}AP_{\mathcal B}.$$

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$P^{-1}AP$

$A$

Basis matrix $P$ by columns

test $[x]_B$

Algorithmic summary

1. Build $P$

Place basis vectors as columns.

2. Convert coordinates

Use $x=P[x]_B$ and $[x]_B=P^{-1}x$.

3. Change matrices

Use $A_B=P^{-1}AP$ for a square transformation.

Quick check

If $P_B$ has basis vectors as columns, what is $P_Bc$?

Common mistake: reversing the direction. $P_B$ goes from $B$-coordinates to standard coordinates; $P_B^{-1}$ goes the other way.