# A calculus of the absurd

#### 22.4 Linear transformations

##### 22.4.1 Introduction to linear transformations
• Definition 22.4.1 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 $$x, y \in \textsf {V}$$,

\begin{align} & T (x + y) = T(x) + T(y) \label {linear transformation homomorphism} \\ & T (\alpha x) = \alpha T(x) \label {linear transformation distributes} \end{align}

• Theorem 22.4.1 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

\begin{equation} T(\alpha x + y) = \alpha T(x) + T(y) \label {simpler linear condition} \end{equation}

Proof: there are two directions to show.

• 1. Only if. We assume that $$T$$ is a linear transformation. Therefore, $$T$$ satisfies 22.106, so we can write

\begin{align} T(\alpha x + y) &= T(\alpha x) + T(y) \\ \end{align}

As $$T$$ is linear it also satisifies 22.107, so by this property,

\begin{align} T(\alpha x + y) &= \alpha T(x) + T(y) \end{align}

• 2. If. We assume that 22.108 holds, and therefore (this is just a restatement of the equation from the theorem)

\begin{align} T(\alpha x + y) = T(\alpha x) + T(y) \end{align} We then set $$\alpha = 1$$, so it follows that

\begin{equation} T(x + y) = T(x) + T(y) \end{equation}

What remains to show is that for all $$\alpha$$ and $$x$$ we have

\begin{equation} T(\alpha x) = \alpha T(x) \end{equation}

We obtain this by fixing $$y = 0$$ from which the result for all $$\alpha$$ and $$\mathbb {K}$$ follows.