Matrix Representations

Level 1 - Math I (Physics) topic page in Linear Maps.

Principle

A matrix representation is a coordinate version of a linear map. Once bases are chosen, applying the map becomes multiplying by a matrix.

Matrix action

\[[T(\mathbf v)]_{\beta}=A[\mathbf v]_{\alpha}\]

The columns of a standard matrix are the images of the standard basis vectors. This makes the whole map recoverable from what it does to a basis.

\(A\)

matrix representing the linear map

\(\mathbf e_i\)

ith standard basis vector

\([\mathbf v]_{\alpha}\)

coordinates of v in input basis alpha

\([T(\mathbf v)]_{\beta}\)

coordinates of the output in basis beta

\(A_{ij}\)

entry in row i and column j

The same physical map can have different matrices in different coordinate systems. The matrix depends on the chosen bases.

In standard coordinates, use \(\\mathbf e_1,\\ldots,\\mathbf e_n\) for the input basis.

Compute \(T(\\mathbf e_1),T(\\mathbf e_2),\\ldots,T(\\mathbf e_n)\).

The coordinate column of \(T(\\mathbf e_j)\) becomes column \(j\) of \(A\).

Column rule

\[A=\bigl[T(\mathbf e_1)\;T(\mathbf e_2)\;\cdots\;T(\mathbf e_n)\bigr]\]

Linear combination

\[T(x_1\mathbf e_1+\cdots+x_n\mathbf e_n)=x_1T(\mathbf e_1)+\cdots+x_nT(\mathbf e_n)\]

The number of columns equals the input dimension. The number of rows equals the output dimension.

Question

Find the standard matrix for

\[T(x,y)=(3x+y,2y)\]

Answer

Compute

\[T(1,0)=(3,0)\]

and

\[T(0,1)=(1,2)\]

These are the columns, so

\[A=\begin{pmatrix}3&1\\0&2\end{pmatrix}\]

Columns come from images of input basis vectors.
Matrix size is output dimension by input dimension.
Changing basis changes the representing matrix.
Do not put basis-vector images into rows unless a convention explicitly says so.