# matrix representation of a linear transformation

## Linear transformations as matrices

Let $V,W$ be vector spaces (over a common field $k$) of dimension  $n$ and $m$ respectively. Fix bases $A=\{v_{1},\ldots,v_{n}\}$ and $B=\{w_{1},\ldots,w_{m}\}$ for $V$ and $W$ respectively. We shall order these bases so that $v_{i} and $w_{i} whenever $i. To distinguish an ordinary set from an ordered set, we shall adopt the notation $\langle v_{1},\ldots,v_{n}\rangle$ to mean the set $\{v_{1},\ldots,v_{n}\}$ with ordering $v_{i}\leq v_{j}$ whenever $i\leq j$. The importance of ordering these bases will be apparent shortly.

For any linear transformation $T:V\to W$, we can write

 $T(v_{j})=\sum_{i=1}^{m}\alpha_{ij}w_{i}$

for each $j\in\{1,\ldots,n\}$ and $\alpha_{ij}\in k$. We define the matrix associated with the linear transformation $T$ and ordered bases $A,B$ by

 $[T]^{A}_{B}:=(\alpha_{ij}),$

where $1\leq i\leq n$ and $1\leq j\leq m$. $[T]^{A}_{B}$ is a $m\times n$ matrix whose entries are in $k$. When $A=B$, we often write $[T]_{A}:=[T]^{A}_{A}$. In addition  , when both ordered bases are standard bases $E_{n},E_{m}$ ordered in the obvious way, we write $[T]:=[T]^{E_{n}}_{E_{m}}$.

Examples.

1. 1.

Let $T:\mathbb{R}^{3}\to\mathbb{R}^{4}$ be given by

 $T\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}x+2y+z\\ z\\ -x+y-5z\\ 3x+2z\end{pmatrix}.$

Using the standard ordered bases

 $E_{3}=\Bigg{\langle}\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\Bigg{\rangle}\mbox{ for }\mathbb{R}^{3}\quad\mbox{ and }\quad E% _{4}=\Bigg{\langle}\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix}\Bigg{\rangle}\mbox{ for }\mathbb{R}^{4}$

ordered in the obvious way. Then,

 $T\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}=\begin{pmatrix}1\\ 0\\ -1\\ 3\end{pmatrix},\quad T\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}=\begin{pmatrix}2\\ 0\\ 1\\ 0\end{pmatrix},\quad T\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}=\begin{pmatrix}1\\ 1\\ -5\\ 2\end{pmatrix},$

so the matrix $[T]^{E_{3}}_{E_{4}}$ associated with $T$ and the standard ordered bases $E_{3}$ and $E_{4}$ is the $4\times 3$ matrix

 $\begin{pmatrix}1&2&1\\ 0&0&1\\ -1&1&-5\\ 3&0&2\end{pmatrix}.$
2. 2.

Let $T$ be the same linear transformation as above. However, let $E^{\prime}_{3}$ be the same basis as $E_{3}$ except that the order is reversed: $e_{3}. Then

 $[T]^{E^{\prime}_{3}}_{E_{4}}=\begin{pmatrix}1&2&1\\ 1&0&0\\ -5&1&-1\\ 2&0&3\end{pmatrix}.$

Note that this matrix is just the matrix from the previous example except that the first and the last columns have been switched.

3. 3.

Again, let $T$ be the same as before. Now, let $E^{\prime}_{4}$ be the ordered basis whose elements are those of $E_{4}$ but the order is now given by $e_{2}. Then

 $[T]^{E^{\prime}_{3}}_{E^{\prime}_{4}}=\begin{pmatrix}1&0&0\\ 1&2&1\\ 2&0&3\\ -5&1&-1\end{pmatrix}.$

Note that this matrix is just the matrix from the previous example except that the first two rows and the last two rows have been interchanged.

Remarks.

## Matrices as linear transformations

Every $m\times n$ matrix $A$ over a field $k$ can be thought of as a linear transformation from $k^{n}$ to $k^{m}$ if we view each vector $v\in k^{n}$ as a $n\times 1$ matrix (a column) and the mapping is done by the matrix multiplication  $Av$, which is a $m\times 1$ matrix (a column vector in $k^{m}$). Specifically, we define $T_{A}:k^{n}\to k^{m}$ by

 $T_{A}(v):=Av.$

It is easy to see that $T_{A}$ is indeed a linear transformation. Furthermore, $[T_{A}]=[T_{A}]^{E_{n}}_{E_{m}}=A$, since the representation of vectors as $n$-tuples of elements in $k$ is the same as expressing each vector under the standard basis (ordered) in the vector space $k^{n}$. Below we list some of the basic properties:

1. 1.

$T_{rA}=rT_{A}$, for any $r\in k$,

2. 2.

$T_{A}+T_{B}=T_{A+B}$, where $A,B$ are $m\times n$ matrices over $k$

3. 3.

$T_{A}\circ T_{B}=T_{AB}$, where $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix over $k$

4. 4.

$T_{A}$ is invertible iff $A$ is an invertible matrix.

Remark. As we can see from the discussion above, if we fix sets of base elements for a vector space $V$ and $W$, there is a one-to-one correspondence between the set of matrices (of the same size) over the underlying field $k$ and the set of linear transformations from $V$ to $W$.

## References

• 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
 Title matrix representation of a linear transformation Canonical name MatrixRepresentationOfALinearTransformation Date of creation 2013-03-22 17:29:59 Last modified on 2013-03-22 17:29:59 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 15 Author CWoo (3771) Entry type Definition Classification msc 15A04 Synonym ordered bases Synonym standard ordered bases Related topic LinearTransformation Defines ordered basis Defines matrix representation Defines standard ordered basis