# Coordinates and linear maps under a change of basis

Aug-Nov 2020

## Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• $m\times n$ matrix A represents a linear map $T:F^n\to F^m$
• dim null $T$ + dim range $T$ = dim $V$
• Linear equation: $Ax=b$
• Solution (if it exists): $u+$ null$(A)$
• Linear map $T$ induces a one-to-one map $V/\text{null }T\to \text{range }T$
• Four fundamental subspaces of a matrix
• Column space, row space, null space, left null space
• Determinant of a square matrix
• Function with many interesting properties

## Coordinates of a vector under a change of basis

coordinates of a vector w.r.t. a basis

• Basis for $V$: $B=\{v_1,\ldots,v_n\}$

• $v\in V$ written as $v=a_1v_1+\cdots+a_nv_n$

• $v\leftrightarrow(a_1,\ldots,a_n)$ w.r.t. basis $B$

change of basis

• Another basis for $V$: $B'=\{v'_1,\ldots,v'_n\}$

• $v_i\in B$ written as $v_i=b_{i1}v'_1+\cdots+b_{in}v'_n$

• Coordinates of $v$ w.r.t. $B'$

$v\leftrightarrow\begin{bmatrix} b_{11}\\\vdots\\b_{1n} \end{bmatrix}a_1+\cdots+\begin{bmatrix} b_{n1}\\\vdots\\b_{nn} \end{bmatrix}a_n=\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}\begin{bmatrix} a_1\\\vdots\\a_n \end{bmatrix}$

any invertible matrix: represents a change of basis

## Change of basis and back

$v\leftrightarrow(a_1,\ldots,a_n)$ w.r.t. basis $B=\{v_1,\ldots,v_n\}$

$v\leftrightarrow\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}\begin{bmatrix} a_1\\\vdots\\a_n \end{bmatrix}$ w.r.t. basis $B'=\{v'_1,\ldots,v'_n\}$

$v'\leftrightarrow(a'_1,\ldots,a'_n)$ w.r.t. basis $B'=\{v'_1,\ldots,v'_n\}$

$v'\leftrightarrow\begin{bmatrix} b'_{11}&\cdots&b'_{n1}\\ \vdots&\vdots&\vdots\\ b'_{1n}&\cdots&b'_{nn} \end{bmatrix}\begin{bmatrix} a'_1\\\vdots\\a'_n \end{bmatrix}$ w.r.t. basis $B=\{v_1,\ldots,v_n\}$

$\begin{bmatrix} b'_{11}&\cdots&b'_{n1}\\ \vdots&\vdots&\vdots\\ b'_{1n}&\cdots&b'_{nn} \end{bmatrix}=\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}^{-1}$

## Matrix of a linear map w.r.t. a basis

$T:V\to W$, linear map

• Basis for $V$: $B_V=\{v_1,\ldots,v_n\}$, Basis for $W$: $B_W=\{w_1,\ldots,w_m\}$

Matrix for $T$ w.r.t. $B_V,B_W$: $\begin{bmatrix} \vdots&\cdots&\vdots\\ T(v_1)&\cdots&T(v_n)\\ \vdots&\cdots&\vdots \end{bmatrix}$

• $i$-th column: coordinates of $T(v_i)$ w.r.t. $B_W$
• $T(v_i)=a_{1i}w_1+\cdots+a_{mi}w_m$

Computing coordinates of $T(v)$

• $v=b_1v_1+\cdots+b_nv_n$

• $T(v)=b_1T(v_1)+\cdots+b_nT(v_n)$

• Coordinates of $T(v)$: $\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ \vdots&\vdots&\vdots\\ a_{m1}&\cdots&a_{mn} \end{bmatrix}\begin{bmatrix} b_1\\\vdots\\b_n \end{bmatrix}$

## Example: Identity map

$I:V\to V$, identity map: $I(v)=v$ for all $v\in V$

• Input basis: $B_V=\{v_1,\ldots,v_n\}$, Output basis: $B_V=\{v_1,\ldots,v_n\}$

Matrix of $I$

• $i$-th column: $I(v_i)=v_i$ w.r.t. $B_V$

$I_n=\begin{bmatrix} 1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1 \end{bmatrix}$, identity matrix

• Input basis: $B_V=\{v_1,\ldots,v_n\}$, Output basis: $B'_V=\{v'_1,\ldots,v'_n\}$

Matrix of $I$

• $i$-th column: $I(v_i)=v_i$ w.r.t. $B'_V$

$\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}$

Identity map w.r.t. $B_V$, $B'_V$: coordinates under change of basis

## Identity map: change of basis and back

Matrix of $I$

• Input basis: $B_V=\{v_1,\ldots,v_n\}$, Output basis: $B'_V=\{v'_1,\ldots,v'_n\}$

• $i$-th column: $I(v_i)=v_i$ w.r.t. $B'_V$

$S=\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}$

Matrix of $I$

• Input basis: $B'_V=\{v'_1,\ldots,v'_n\}$, Output basis: $B_V=\{v_1,\ldots,v_n\}$

• $i$-th column: $I(v'_i)=v'_i$ w.r.t. $B_V$

$S^{-1}=\begin{bmatrix} b_{11}&\cdots&b_{n1}\\ \vdots&\vdots&\vdots\\ b_{1n}&\cdots&b_{nn} \end{bmatrix}^{-1}$

## Linear maps and change of basis

$T:V\to W$, linear map

• Basis for $V$: $B_V=\{v_1,\ldots,v_n\}$, Basis for $W$: $B_W=\{w_1,\ldots,w_m\}$

Matrix for $T$ w.r.t. $B_V,B_W$: $\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ \vdots&\vdots&\vdots\\ a_{m1}&\cdots&a_{mn} \end{bmatrix}$

Change of basis

• Basis for $V$: $B'_V=\{v'_1,\ldots,v'_n\}$, Basis for $W$: $B'_W=\{w'_1,\ldots,w'_m\}$

view as composition

$I(B_W\to B'_W)\ T(B_V\to B_W)\ I(B'_V\to B_V)$

Matrix: (matrix for $I$ in W) (matrix for $T$) (matrix for $I$ in V)

## Change of basis for operators

$T:V\to V$, linear operator

• Input basis: $B_V=\{v_1,\ldots,v_n\}$, Output basis: $B_V=\{v_1,\ldots,v_n\}$

Matrix for $T$: $A=\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ \vdots&\vdots&\vdots\\ a_{n1}&\cdots&a_{nn} \end{bmatrix}$

change of basis

• Input basis: $B'_V=\{v'_1,\ldots,v'_n\}$, Output basis: $B'_V=\{v'_1,\ldots,v'_n\}$

Matrix for $T$: $S\ A\ S^{-1}$

similarity transform

$S$: any invertible matrix

$SAS^{-1}$: represents change of basis to $\{$columns of $S\}$