Linear Tranformation

Chentian Wu

2024-11-09

MATH › Linear Algebra

\newcommand{\0}{\textbf 0} \newcommand{\1}{\textbf 1} \newcommand{\x}{\boldsymbol x} \newcommand{\y}{\boldsymbol y} \newcommand{\z}{\boldsymbol z} \newcommand{\u}{\boldsymbol u} \newcommand{\v}{\boldsymbol v} \newcommand{\w}{\boldsymbol w} \newcommand{\L}{\mathcal L} \newcommand{\vn}{\{ {\boldsymbol v}_1, {\boldsymbol v}_2,\ldots,{\boldsymbol v}_n\} } \newcommand{\wn}{\{ {\boldsymbol w}_1, {\boldsymbol w}_2,\ldots,{\boldsymbol w}_n\} } \newcommand{\Ker}{\text{ker}} \renewcommand{\Im}{\text{Im}} \newcommand{\Span}{\text{span}} \newcommand{\Norm}{\unlhd} \newcommand{\Null}{\text{nullity}} \newcommand{\Rank}{\text{rank}} \newcommand{\Hom}{\text{Hom}}

1. Linear Transformation

Definition. Linear transformation

Let $$V$$ and $$W$$ be vector spaces (over $$F$$ ). We call a function $T:V\to W$ a linear transformation from $$V$$ to $$W$$ if $\forall \x,\y \in V,\forall c\in F$ .

\begin{aligned} T(\x+\y)&=T(\x)+T(\y)\\ T(c\x)&=cT(\x) \end{aligned}

If $$T$$ is a linear transformation, we often just call $$T$$ linear. It could be checked that $$T$$ is linear iff for any $\{\x_k\}_{k=1}^n$ and $\{a_k\}_{k=1}^n$ , we have

T(\sum_{k=1}^n a_k\x_k)=\sum_{k=1}^na_kT(\x_k)

By definition, a linear transformation is just a module homomorphism $$T$$ in $\Hom_F(V,W)$

Definition. Identity transformation
For vector space $$V$$ over $$F$$ , the identity transformation $$I_V$$ is defined as follows

\begin{aligned} I_V:V&\to V\\ \v&\mapsto \v \end{aligned}

We can write $$I$$ instead of $$I_V$$

Definition. Zero transformation
For vector space $$V$$ and $$W$$ over $$F$$ , the zero transformation $$T_0$$ is defined as follows

\begin{aligned} T_0:V&\to W\\ \x&\mapsto \0 \end{aligned}

Definition. Null space / kernel, range
Let $$V$$ and $$W$$ be two vector spaces. $T:V\to W$ is linear. We define the null space $$N(T)$$ / kernel $\Ker(T)$ of $$T$$ as

\Ker(T):=\{\x\in V\mid T(\x)=\0\}=:T^{-1}(\{\0\})

The range / image of $$T$$ is defined as $\Im(T):=\{T(\v)\mid \v\in V\}=:T(V)$

Theorem. Let $$V$$ and $$W$$ be vector spaces and $T:V\to W$ is linear. Then $\Ker(T)$ is a subspace of $$V$$ , and $\Im(T)$ is a subspace of $$W$$ .
Pf. Easy to show using definition of subspaces.

From the perspective of groups, $$N$$ is the kernel of some group homomorphism from $$G$$ is equivalent to $N\Norm G$ . Here $$V$$ is abelian, so every kernel should be corresponding a subspace of $$V$$ .

****Theorem**. Let $$V$$ and $$W$$ be vector spaces and $T:V\to W$ is linear. If $\beta=\{\v_k\}_{k=1}^n$ is a basis for $$V$$ , then

\Im(T)=\Span(T(\beta))=\Span(\{T(\v_1), T(\v_2), \ldots, T(\v_n)\})

Definition. Nullity / rank
Let $$V$$ and $$W$$ be two vector spaces. $T:V\to W$ is linear. We define the nullity (denoted by $\Null(T)$ ) and rank of $$T$$ (denoted by $\Rank(T)$ ) as follows

\begin{aligned} \Null(T)&:=\dim(\Ker(T))\\ \Rank(T)&:=\dim(\Im(T)) \end{aligned}

Theorem. Dimension theorem
Let $$V$$ and $$W$$ be vector spaces, let $T:V\to W$ be linear. If $$V$$ is finite-dimensional, then

\Null(T)+\Rank(T)=\dim(V)

Pf.

Lemma. Properties of linear transformations between equal-dimensional vector spaces
Let $$V$$ and $$W$$ be vector spaces with equal (finite) dimensions, and let $T:V\to W$ be linear. TTFE

$$T$$ is surjective
$$T$$ is injective
$\Rank(T)=\dim(V)$
Pf. Firstly note that $$T$$ is injective iff. $\Ker(T)=\{\0\}$ because

T(\x)=T(\y)\iff T(\x-\y)=\0\iff (\x-\y)\in\Ker(T)

So by Dimension Theorem 1. and 3. are equivalent. Note that $T(V)=\Im(T)$ is a subspace of $$W$$ , so

\Rank(T)=\dim(V)\iff \dim(\Im(T))=\dim(W)=\dim(V)\iff \Im(T)=W

that is, if a space have a subspace with the same dimension, then they are equal.

Theorem. Uniqueness of linear transformation between same dimensional vector spaces
Let $$V$$ and $$W$$ be vector spaces over $$F$$ . Suppose $\vn$ is a basis for $$V$$ . For any $\wn$ , there exists a unique linear transformation $T:V\to W$ such that

\forall i\in\{1,2,\ldots,n\}.T(\v_i)=\w_i

Definition. Invariant
Let $$V$$ be a vector space, $T:V\to V$ is linear. A subspace $$W$$ of $$V$$ is said to be $$T$$ -invariant if

\forall x\in W. T(x)\in W

that is $T(W)\subset V$

2. Matrix Representation

Definition. Ordered basis
An ordered basis is nothing more than a basis endowed with a specific order.

We need to fix the order of basis so that we can better represent a transformation as a matrix.

Definition. Coordinate vector
Let $\beta=\{\u_1,\u_2,\ldots,\u_n\}$ be an ordered basis for a finite dimensional vector space $$V$$ . For $x\in V$ , let $a_1, a_2,\ldots, a_n$ be the unique scalars such that

x=\sum_{i=1}^na_i\u_i

We define the coordinate vector of $$x$$ relative to $\beta$ , denoted as $[x]_\beta$ , by

[x]_\beta=\begin{pmatrix}a_1\\a_2\\\vdots \\ a_n\end{pmatrix}

Definition. Addition and scalar multiplication of linear transformations
Let $$T, U$$ be linear transformations from $$V$$ to $$W$$ where $$V$$ and $$F$$ are vector spaces over $$F$$ . Let $a\in F$ , $x\in V$ be an arbitrary vector.
Define $T+U:V\to W$ as

T+W:x\mapsto T(x)+U(x)

and define $$aT$$ as

aT:x\mapsto a(T(x))

Theorem. The collection of all linear transformations is a vector space
Let $\L(V,W)$ denote the collection of all linear transformations from $$V$$ to $$W$$ where $$V$$ and $$W$$ are vector spaces over $$F$$ . Then $\L(V, W)$ is a vector space over $$F$$ .

Theorem. A composition of linear transformations is still linear

Definition. Invertible transformation

Definition. Standard representation

Definition. Similar matrices
Let $$A$$ be $$B$$ be matrices in $M_{n\times n}(F)$ . We say that $$A$$ is similar to $$B$$ if there exists an invertible matrix $$Q$$ such that $A=Q^{-1}BQ$

the relation of “is similar to” is obviously an equivalence relation.