Linear maps

Overview §

Linear maps are functions that send elements from one vector space to another in a way that is compatible with the Vector space and subspace axioms.

Linearity is ubiquitous, meaning addition and scaling can be applied before or after the linear transformation without changing the result.

Linear maps are defined with respect to a basis. One upshot of this fact is that equality can be checked with a finite list of vectors.

---

Linearity §

Definition: Linear map

If $V,W$ are vector spaces over the field $\mathbb{F}$, a linear map (i.e., linear transformation, linear operator) from $V$ to $W$ is a function $A:V\to W$ with the following properties:

• (L1) $A$ respects addition. For all $v_1,v_2\in V$, $A(v_1+v_2)=A(v_1)+A(v_2)$
• (L2) $A$ respects scaling. For all $v\in V$ and $c\in \mathbb{F}$, $A(cv)=cA(v)$.

Every linear function $f:{\mathbb{F}}^n\to F$ for some $a_1,\cdots ,a_n\in \mathbb{F}$ takes the form

$$f\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=a_1{x}_1+\cdots +a_n{x}_n.$$

Properties of linear maps include:

• (Lemma 44) If $A:V\to W$ is a linear map, we have $A(\vec{0})=\vec{0}.$
  • Informally, this means linear maps do not have constant terms. A function in the form $f(x)=ax+b$ for $b\ne 0$ is an affine function.
• (Theorem) Linear maps are determined by their values on a basis

---

Examples of linear maps §

• Rotation
• Differentiation and integration
• Identity map

#wip Make notes for each of these examples