04 - Linear Maps

Linear Map / Transformation

Suppose $V$ and $W$ are Vector Spaces over $\mathbf{F}$

Map $T:V\to W$ is a linear transformation (or linear map)
If for all $a\in \mathbf{F}$ and $v,v^{\prime }\in V$ then

$$T(av+v^{\prime })=aT(v)+T(v^{\prime })$$

Isomorphism of Vector Spaces

Bijective linear map of vector spaces

Homomorphism between Vector Spaces

Let $V,W$ be vector spaces then
Define $\mathrm{Hom}(V,W)$ be the set of linear maps from $V$ to $W$

For $a\in \mathbf{F}$ and $v\in V$ and $S,T\in \mathrm{Hom}(V,W)$ define

$$\begin{array}{rl}(aT)(v)& :=a(T(v))\\ (T+S)(v)& :=T(v)+S(v)\end{array}$$

Then $\mathrm{Hom}(V,W)$ is a vector space over $\mathbf{F}$

Linear Maps are determined by a basis

Let $V,W$ be finite dimensional vector spaces

Every linear map $T:V\to W$ is determined by its values on a basis $\mathcal{B}$ for $V$ (as $\mathcal{B}$ is spanning)

and vice versa so

Any map $T:\mathcal{B}\to W$ can be extended to linear map $T:V\to W$ (as $\mathcal{B}$ is linearly independent)

Matrix Representation of a Linear Map

Let $\mathcal{B}=\{e_1,\cdots ,e_n\}$ and ${\mathcal{B}}^{\prime }=\{e_1^{\prime },\cdots ,e_m^{\prime }\}$
Let ${}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}$ be the matrix with $(i,j)$-entry $a_{ij}$ such that

$$T(e_j)=a_{1j}{e}_1^{\prime }+\cdots +a_{mj}{e}_m^{\prime }$$

where $\mathcal{B}$ is the initial basis and ${\mathcal{B}}^{\prime }$ is the final basis

Properties of Matrix Representation of a Linear Map

1. Scalar Multiplication

$${}_{{\mathcal{B}}^{\prime }}[aT{]}_{\mathcal{B}}=a({}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}})$$

2. Addition

$${}_{{\mathcal{B}}^{\prime }}[T+S{]}_{\mathcal{B}}={}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}+{}_{{\mathcal{B}}^{\prime }}[S{]}_{\mathcal{B}}$$

3. Composition

$${}_{{\mathcal{B}}^{\prime \prime }}[S\circ T{]}_{\mathcal{B}}={}_{{\mathcal{B}}^{\prime \prime }}[S{]}_{{\mathcal{B}}^{\prime }}{}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}$$

---

Nilpotent

Let $V$ be finite-dimensional
Let $T:V\to V$ be a linear transformation

If

$$T^n=0\text{ for some }n>0$$

Then

$$T\text{ is nilpotent}$$