## Section2Linear transformations

### Subsection2.1Definition of a linear transformation

A function $$L\colon \R^n\to \R^m$$ is called a linear transformation (or a linear mapping, or simply a linear map) if

1. $$L(\mathbf{x}+\mathbf{y}) = L(\mathbf{x}) + L(\mathbf{y})\text{,}$$ and
2. $$\displaystyle L(\alpha\mathbf{x}) = \alpha L(\mathbf{x})$$

for all vectors $$\mathbf{x},\mathbf{y}$$ in $$\R^n$$ and real numbers $$\alpha\text{.}$$ Properties (i) and (ii) are called linearity properties. We say that $$L$$ preserves or respects vector operations of addition and scaling. Instead of $$L(\mathbf{x})\text{,}$$ it is common practice to omit the parentheses and write $$L\mathbf{x}$$ when $$L$$ is a linear transformation. For the case when $$m=n\text{,}$$ a linear transformation $$L\colon \R^n\to \R^n$$ is also called a linear operator on $$\R^n\text{.}$$

### Subsection2.2Why linear maps are easy to compute

Given a vector $$\mathbf{x}=(x_1,\ldots,x_n)$$ and a linear map $$L\colon \R^n\to\R^m\text{,}$$ we have

\begin{align} L\mathbf{x} \amp = L\left(\sum_{j=1}^n x_j\mathbf{e}_j\right)\label{linmapsumstd1}\tag{2.1}\\ \amp = \sum_{j=1}^n L(x_j\mathbf{e}_j)\label{linmapsumstd2}\tag{2.2}\\ \amp = \sum_{j=1}^n x_j L\mathbf{e}_j\label{linmapsumstd3}\tag{2.3} \end{align}

A consequence of equation

\begin{equation} L\mathbf{x} = \sum_j x_j L\mathbf{e}_j\label{linmapsumstdfact}\tag{2.4} \end{equation}

is that a linear map is determined by its values on the standard basis vectors $$\mathbf{e}_1, \mathbf{e}_2,\ldots,\mathbf{e}_n\text{.}$$ In words, the value of $$L$$ on the vector $$\mathbf{x}$$ is a linear combination of the vectors $$L\mathbf{e}_j\text{,}$$ with the coefficient $$x_j$$ for the vector $$L\mathbf{e}_j\text{.}$$ Given the vectors $$L\mathbf{e}_j\text{,}$$ computing the value of a linear map requires only scalar multiplication and vector addition.

### Subsection2.3Operations on linear transformations

Let $$L, L'\colon \R^n\to \R^m\text{,}$$ let $$M\colon \R^p\to \R^n\text{,}$$ and let $$\alpha$$ be a real number. Linear transformations $$\alpha L\colon \R^n\to \R^m\text{,}$$ $$L+L'\colon \R^n \to \R^m\text{,}$$ and $$LM\colon \R^p\to \R^m$$ are defined as follows.

\begin{align} (\alpha L)\mathbf{x} \amp = \alpha (L\mathbf{x})\amp \text{(scalar multiplication)}\label{linmapscale}\tag{2.5}\\ (L+L')\mathbf{x} \amp = L\mathbf{x} + L'\mathbf{x}\amp \text{(addition)}\label{linmapadd}\tag{2.6}\\ (LM)\mathbf{x} \amp = L(M\mathbf{x})\amp \text{(composition)}\label{linmapcompose}\tag{2.7} \end{align}

Note that $$LM$$ is the same thing as $$L\circ M\text{,}$$ the ordinary composition of functions. It is conventional to omit the composition symbol in the context of linear transformations.

### Exercises2.4Exercises

###### 1.

Let $$L\colon \R^2\to\R^2$$ be a linear map such that $$L\mathbf{e}_1=(2,3)$$ and $$L\mathbf{e}_2=(-1,-2)\text{.}$$ Find $$L(1,2)\text{.}$$

$$L(1,2)=(0,-1)$$

###### 2.

Let $$L\colon \R^3\to\R$$ be a linear map. Find $$L{\mathbf k}$$ if $$L{\mathbf i}=2\text{,}$$ $$L{\mathbf j}=-1\text{,}$$ and $$L(1,2,3)=0\text{.}$$

$$L{\mathbf k} = 0$$

###### 3.

Show that the two linearity properties in the definition of linear transformation given in Subsection 2.1 are equivalent to the single property

\begin{equation} L(a\mathbf{x}+b\mathbf{y}) = a L\mathbf{x} + bL\mathbf{y}\label{linearityprop1eqn}\tag{2.8} \end{equation}

for all vectors $$\mathbf{x},\mathbf{y}$$ and scalars $$a,b\text{.}$$

###### 4.

Justify each of the equalities (2.1), (2.2), and (2.3).

###### 5.

The dot product has the following properties that look like the properties in the definition of linear map.

\begin{align} \mathbf{u}\cdot (\mathbf{v}+\mathbf{w}) \amp = \mathbf{u}\cdot \mathbf{v} + \mathbf{u}\cdot \mathbf{w}\label{dotprodlinprop1}\tag{2.9}\\ \mathbf{u}\cdot (\alpha \mathbf{v}) \amp = (\alpha \mathbf{u})\cdot \mathbf{v}= \alpha (\mathbf{u}\cdot\mathbf{v})\label{dotprodlinprop2}\tag{2.10} \end{align}

for all $$\mathbf{u},\mathbf{v},\mathbf{w}$$ in $$\R^n$$ and scalars $$\alpha\text{.}$$ Show that these properties hold.

###### 6.

Prove that the composition of two linear maps is a linear map.