Chapter 1 extra note 3

Linear transformation
Matrix representation of a linear transformation
Range
kernel

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Linear transformation

Definition

Let

$V$,

$W$ be vector spaces over the same field

$F$. The map

$T:V\to W$ is a linear transformation if

1. $T(u+v)=T(u)+T(v),\mathrm{\forall }u,v\in V$.
2. $T(\alpha v)=\alpha T(v),\mathrm{\forall }\alpha \in F,v\in V$.

Or equivalently,

$$T(\alpha u+\beta v)=\alpha T(u)+\beta T(v),\mathrm{\forall }\alpha ,\beta \in F,u,v\in V.$$

Lemma:

Let

$T:V\to W$ be a linear transformation, then

$T(\mathbf{0})=\mathbf{0}$.

• Proof:

  Let

  $\mathbf{u}\in V$, then
  

  $$T(\mathbf{0})=T(0\cdot \mathbf{u})=0\cdot T(\mathbf{u})=\mathbf{0}.$$

Examples of linear transformation

• $T:{\mathbb{P}}_2(x)\to {\mathbb{P}}_1(x)$ with
  $T(p)=\frac{d}{dx}p(x)$.
• $T:{\mathbb{P}}_n(x)\to {\mathbb{P}}_{n+1}(x)$ with
  $T(p)={\int }_0^{x}p(s)\text{d}s$.
• $T:{\mathbb{P}}_3(x)\to {\mathbb{P}}_3(x)$ with
  $T(p)=\frac{d}{dx}p(x)$.
  • a) Let
    $\beta =\{1,x,x^2,x^3\}$ be a basis of
    ${\mathbb{P}}_3(x)$. For any
    $\mathbf{v}\in {\mathbb{P}}_3(x)$, we have
    
    $$\mathbf{v}={\alpha }_0\cdot 1+{\alpha }_1\cdot x+{\alpha }_2\cdot x^2+{\alpha }_3\cdot x^3.$$
    Also, we have
    
    $$\begin{array}{}\text{(2)}& T(\mathbf{v})={\alpha }_0\cdot T(1)+{\alpha }_1\cdot T(x)+{\alpha }_2\cdot T(x^2)+{\alpha }_3\cdot T(x^3),\end{array}$$
    and notice that
    
    $$\begin{array}{rl}T(1)& =0=0\cdot 1+0\cdot x+0\cdot x^2+0\cdot x^3=\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\\ T(x)& =1=\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],\\ T(x^2)& =2x=\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{c}0\\ 2\\ 0\\ 0\end{array}\right],\\ T(x^3)& =3x^2=\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 3\\ 0\end{array}\right].\end{array}$$
    We can then rewrite (2) as
    
    $$\begin{array}{rl}T(\mathbf{v})& =\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]({\alpha }_0\cdot \left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]+{\alpha }_1\cdot \left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]+{\alpha }_2\cdot \left[\begin{array}{c}0\\ 2\\ 0\\ 0\end{array}\right]+{\alpha }_3\cdot \left[\begin{array}{c}0\\ 0\\ 3\\ 0\end{array}\right])\\ & =\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 3\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\alpha }_0\\ {\alpha }_1\\ {\alpha }_2\\ {\alpha }_3\end{array}\right]\\ & =\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ 2{\alpha }_2\\ 3{\alpha }_3\\ 0\end{array}\right]\\ & =\alpha \cdot 1+(2\alpha 2)\cdot x+(3{\alpha }_3)\cdot x^2+0\cdot x^3.\end{array}$$
    Therefore, given coordinates of
    $\mathbf{v}$,
    $({\alpha }_1,\cdots ,{\alpha }_4)$, it is clear that the following matrix-vector multiplication gives the coordinates of
    $T(\mathbf{v})$:
    
    $$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 3\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\alpha }_0\\ {\alpha }_1\\ {\alpha }_2\\ {\alpha }_3\end{array}\right].$$
  • b) Let
    $\gamma =\{1,2x,3x^2,4x^3\}$ be another basis of
    ${\mathbb{P}}_3(x)$, and we now write
    $T(\mathbf{v})$ in the basis
    $\gamma$. We have
    
    $$\begin{array}{rl}T(1)& =0=\left[\begin{array}{cccc}1& 2x& 3x^2& 4x^3\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\\ T(x)& =1=\left[\begin{array}{cccc}1& 2x& 3x^2& 4x^3\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],\\ T(x^2)& =2x=\left[\begin{array}{cccc}1& 2x& 3x^2& 4x^3\end{array}\right]\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right],\\ T(x^3)& =3x^2=\left[\begin{array}{cccc}1& 2x& 3x^2& 4x^3\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right],\end{array}$$
    and the matrix representation of
    $T(\mathbf{v})$ is
    
    $$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\alpha }_0\\ {\alpha }_1\\ {\alpha }_2\\ {\alpha }_3\end{array}\right].$$

---

Matrix representation of a linear transformation

Definition:
Let

$\beta =v_1,\cdots ,v_n$ be a basis of

$V$ and

$\gamma =w_1,\cdots ,w_m$ be a basis of

$W$, and let

$T:V\to W$ be a linear transformation. Then, for each

$j$ with

$1\le j\le n$, there exists

$\{a_{ij},1\le i\le m\}$ such that

$$T(v_j)=\sum _{i=1}^m{a}_{ij}{w}_i.$$
The

$m\times n$ matrix

$[a_{ij}]$ is called the matrix representation of

$T$ in basis

$\beta$ and

$\gamma$, and is denoted by

$[T{]}_{\beta }^{\gamma }$.

If

$T:V\to V$ with the same basis

$\beta$, the matrix representation is denoted by

$[T{]}_{\beta }$.

Examples of the matrix representation

• $T:{\mathbb{P}}_3(x)\to {\mathbb{P}}_3(x)$ with
  $T(p)=\frac{d}{dx}p(x)$.
  • $\beta =\{1,x,x^2,x^3\}$ and
    $\gamma =\{1,2x,3x^2,4x^3\}$.
    
    $$[T{]}_{\beta }=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 3\\ 0& 0& 0& 0\end{array}\right],[T{]}_{\beta }^{\gamma }=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right].$$
• $T:{\mathbb{P}}_2(x)\to {\mathbb{P}}_2(x)$ with
  $T(p)=p^{″}-3p^{\prime }+3p$.
  • $\beta =\{1,x,x^2\}$.
    • First approach - We calculate the coefficients of
      $T(1)$,
      $T(x)$ and
      $T(x^2)$ to have
      
      $$[T{]}_{\beta }=\left[\begin{array}{ccc}3& -3& 2\\ 0& 3& -6\\ 0& 0& 3\end{array}\right].$$
    • Second approach - Let's define two linear transformations,
      $D:{\mathbb{P}}_2(x)\to {\mathbb{P}}_2(x)$ with
      $D(p)=p^{\prime }$ and
      $I:{\mathbb{P}}_2(x)\to {\mathbb{P}}_2(x)$ with
      $I(p)=p$. It is then clear that
      
      $$T=D\circ D-3D+3I.$$
      The matrix of
      $D$ and
      $I$ are
      
      $$[D{]}_{\beta }=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 2\\ 0& 0& 0\end{array}\right],[I{]}_{\beta }=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],$$
      therefore
      
      $$[T{]}_{\beta }=D^2-3D+3I=\left[\begin{array}{ccc}3& -3& 2\\ 0& 3& -6\\ 0& 0& 3\end{array}\right].$$

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Range and kernel

Definition:
Given

$T:V\to W$ be a linear transformation.

• The range of
  $T$ is defined as
  
  $$\text{range}(T)=\{\mathbf{w}\in W|T(\mathbf{v})=\mathbf{w},\mathbf{v}\in V\}.$$
  • Other names:
    
    $$\text{range}(T)=\text{Ran}(T)=\text{image}(T)=\text{Im}(T).$$
• The kernel of
  $T$ is defined as
  
  $$\text{kernel}(T)=\{\mathbf{v}\in V|T(\mathbf{v})=\mathbf{0}\}.$$
  • Other names:
    
    $$\text{kernel}(T)=\text{Ker}(T)=\text{null}(T)=\text{N}(T).$$