19.3 Linear transformations
19.3.1 Introduction to linear transformations
Definition 19.3.1
Let $\textsf{V},\textsf{W}$ be vector spaces over a field $\mathbb{K}$. We say a function $T:\textsf{V}\to\textsf{W}$ is linear if it satisfies these two properties

1.
For every $x,y\in\textsf{V}$,
$\displaystyle T(x+y)=T(x)+T(y)$ (19.84) $\displaystyle T(\alpha x)=\alpha T(x)$ (19.85)
Theorem 19.3.1
Let $\textsf{V},\textsf{W}$ be vector spaces over a field $\mathbb{K}$ and let $x,y\in\textsf{V}$ and $\alpha\in\mathbb{K}$. The map/function/ whatever you want to call it $T:\textsf{V}\to\textsf{W}$ is linear if and only if
Proof: there are two directions to show.

1.
Only if. We assume that $T$ is a linear transformation. Therefore, $T$ satisfies 19.84, so we can write
$\displaystyle T(\alpha x+y)$ $\displaystyle=T(\alpha x)+T(y)$ (19.87) As $T$ is linear it also satisifies 19.85, so by this property,
$\displaystyle T(\alpha x+y)$ $\displaystyle=\alpha T(x)+T(y)$ (19.89) 
2.
If. We assume that 19.86 holds, and therefore (this is just a restatement of the equation from the theorem)
$\displaystyle T(\alpha x+y)=T(\alpha x)+T(y)$ (19.90) We then set $\alpha=1$, so it follows that
$T(x+y)=T(x)+T(y)$ (19.91)What remains to show is that for all $\alpha$ and $x$ we have
$T(\alpha x)=\alpha T(x)$ (19.92)We obtain this by fixing $y=0$ from which the result for all $\alpha$ and $\mathbb{K}$ follows.
19.3.2 The matrix representation of a linear transformation
The best way to understand this is to do a lot of examples, with specific linear transformations and vector spaces. It’s easy to get lost, sinking, \saynot waving but drowning in the steaming soup of generality. As they don’t say, a little reification ^{7}^{7} 7 Meaning turning something abstract into something concrete. every day keeps the doctor away.
Let’s assume that we have a linear transformation $T:V\to W$, and we would like to find its matrix representation. It’s really easy to get confused here, but don’t lose sight of the goal. We need some information about $V$ and $W$, specifically

•
A basis for $V$, denoted as $\mathcal{B}=\{\beta_{i}\}_{1\leqq i\leqq n}$.

•
A basis for $W$, denoted as $\mathcal{C}=\{\gamma_{i}\}_{1\leqq i\leqq m}$.
We then pick an arbitrary vector, $\mathbf{v}\in V$, and finds its representation as a linear combination of $\beta$, that is we find
But we’re not after $\mathbf{v}$, we’re after $T(\mathbf{v})$! Therefore, we apply $T$ to both sides, giving
$\displaystyle T(\mathbf{v})$  $\displaystyle=T\left(\sum_{1\leqq j\leqq n}v_{j}\beta_{j}\right)$  (19.94) 
We now use liberally the fact that $T$ is linear.
$\displaystyle T(\mathbf{v})$  $\displaystyle=\sum_{1\leqq j\leqq n}v_{j}T(\beta_{j})$  (19.96) 
We’re not dealing with a concrete linear transformation, so \sayall we can say is that for each $i$, $T(\beta_{j})$ will give us a vector in $W$ and that we can certainly write this as a linear combination of $\mathcal{C}$, as it is a basis for $W$. Every $T(\beta_{j})$ is a linear combination of the $m$ vectors in $\mathcal{C}$, i.e. $T(\beta_{j})=\sum_{1\leqq i\leqq m}\left(a_{i,j}\right)\gamma_{i}$. Substituting this in, we get
$\displaystyle T(\mathbf{v})$  $\displaystyle=\sum_{1\leqq j\leqq n}v_{j}\left(\sum_{1\leqq i\leqq m}a_{i,j}% \gamma_{i}\right)$  (19.97)  
$\displaystyle=\sum_{1\leqq i\leqq m}\gamma_{i}\left(\sum_{1\leqq j\leqq n}a_{i% ,j}v_{j}\right)$  (19.98)  
$\displaystyle=\sum_{1\leqq i\leqq m}\left(\sum_{1\leqq j\leqq n}a_{i,j}v_{j}% \right)\gamma_{i}$  (19.99) 
Now, from the definition of a coordinate vector, as $\gamma_{1},...,\gamma_{m}$ are the basis vectors for $\mathcal{C}$, the representation of $T(v)$ as a coordinate vector in this basis is just
$\displaystyle[T(v)]_{\mathcal{C}}$  $\displaystyle=\begin{pmatrix}\sum_{1\leqq j\leqq n}a_{1,j}v_{j}\\ \sum_{1\leqq j\leqq n}a_{2,j}v_{j}\\ ...\\ \sum_{1\leqq j\leqq n}a_{m,j}v_{j}\\ \end{pmatrix}$  (19.104)  
$\displaystyle=\underbrace{\begin{pmatrix}a_{1,1}&a_{1,2}&...&a_{1,n}\\ a_{2,1}&a_{2,2}&...&a_{2,n}\\ ...&...&...&...\\ a_{m,1}&a_{1,2}&...&a_{1,n}\end{pmatrix}}_{\text{Let this be $\mathbf{A}$.}}% \begin{pmatrix}v_{1}\\ v_{2}\\ ...\\ v_{3}\end{pmatrix}$  (19.113) 
Which is exactly what we wanted to find. Specifically, the $j$th column of the matrix $\mathbf{A}$ (as defined in Equation 19.113) is the coordinate vector (in the ordered base $\mathcal{C}$) of the result of $T(\beta_{j})$.