Coordinates and linear maps under a change of basis

Andrew Thangaraj

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 \(T\): \(S^{-1}\ A\ S\)

similarity transform

\(S\): any invertible matrix

\(S^{-1}AS\): change of basis to \(\{\)columns of \(S\}\)

Quiz