Linear Map

Definition: Linear Map

A linear map from $V$ to $W$ is a function $T:V\to W$ with the following properties.

• Additivity: $T(u+v)=Tu+Tv$ for all $u,v\in V$.
• Homogeneity: $T(\lambda v)=\lambda (Tv)$ for all $\lambda \in \mathbb{F}$ and all $v\in V$.

We use the notation $Tv$ as well as the function notation $T(v)$. Some use the term linear transformation instead.

Notation: $\mathcal{L}(V,W),\mathcal{L}(V)$

• The set of linear maps from $V$ to $W$ is denoted by $\mathcal{L}(V,W)$
• The set of linear maps from $V$ to $V$ is denoted by $\mathcal{L}(V)$. In other words, $\mathcal{L}(V)=\mathcal{L}(V,V)$.

Note that linear functions are not the same as linear maps! Suppose $m,b\in \mathbb{R}$. The function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=mx+b$ is a linear map if and only if $b=0$.

Examples

Zero

We let the symbol $0$ denote the linear map that takes every element of some vector space to the additive identity of another (or possibly the same) vector space. To be specific, $0\in \mathcal{L}(V,W)$ is defined by

$$0v=0$$

• The $0$ on the left side of the equation above is a function from $V$ to $W$
• The $0$ on the right side is the additive identity in $W$

Identity Operator

The identity operator, denoted by $I$, is the linear map on some vector space that takes each element to itself. To be specific, $I\in \mathcal{L}(V)$ is defined by

$$Iv=v$$

Differentiation

Define $D\in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ by

$$Dp=p^{\prime }$$

The assertion that this function is a linear map is another way of stating a basic result about differentiation: $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$ and $(\lambda f{)}^{\prime }=\lambda f^{\prime }$ whenever $f,g$ are differentiable and $\lambda$ is constant.

Integration

Define $T\in \mathcal{L}(\mathcal{P}(\mathbb{R}),\mathbb{R})$ by

$$Tp={\int }_0^{1}p$$

The assertion that this function is linear is another way of stating a basic result about integration: the integration of the sum of two functions equals the sum of the integrals, and the integral of a constant times a function equals the constant times the integral of the function.

Multiplication by $x^2$

Define a linear map $T\in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ by

$$(Tp)(x)=x^{2}p(x)$$

for each $x\in \mathbb{R}$.

Backward Shift

Recall that ${\mathbb{F}}^{\infty }$ denotes the vector space of all sequence of elements of $\mathbb{F}$. Define a linear map $T\in \mathcal{L}({\mathbb{F}}^{\infty })$ by

$$T(x_1,x_2,x_3,\dots )=(x_2,x_3,\dots )$$

From ${\mathbb{R}}^3$ to ${\mathbb{R}}^2$

Define a linear map $T\in \mathcal{L}({\mathbb{R}}^3,{\mathbb{R}}^2)$ by

$$T(x,y,z)=(2x-y+3z,7x+5y-6z)$$

From ${\mathbb{F}}^n$ to ${\mathbb{F}}^m$

To generalize the previous example, let $m$ and $n$ be positive integers, let $A_{j,k}\in \mathbb{F}$ for each $j=1,\dots ,m$ and each $k=1,\dots ,n$, and define a linear map $T\in \mathcal{L}({\mathbb{F}}^n,{\mathbb{F}}^m)$ by

$$T(x_1,\dots ,x_n)=(A_{1,1}{x}_1+\cdots +A_{1,n}{x}_n,\dots ,A_{m,1}{x}_1+\cdots +A_{m,n}{x}_n)$$

Actually every linear map from ${\mathbb{F}}^n$ to ${\mathbb{F}}^m$ is of this form.

Composition

Fix a polynomial $q\in \mathcal{P}(\mathbb{R})$. Define a linear map $T\in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ by

$$(Tp)(x)=p(q(x))$$