3.2 linear mappings

What are linear mappings

• In general a linear mapping is

  • A linear mapping is a function $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ that preserves the linear combinations that is for all $s,t\in \mathbb{R}$:

$$L(s\vec{u}+t\vec{v})=sL(\vec{u})+tL(\vec{v})$$

Range of a linear mapping

• Let L:${\mathbb{R}}^n\to {\mathbb{R}}^m$ be a linear mapping

  • the range of L is the set $⟨L(\vec{x})|\vec{x}\in {\mathbb{R}}^n)$
    • range is also equal to the 3.5 Column Space
  • the kernal (or “nullspace”) of L is set $⟨\vec{x}\in {\mathbb{R}}^n|L(\vec{x})=\vec{0})$
    • denoted Range(L) and ker(L),respectively
      • kernal space is equal to the 3.4Nullspace and left nullspace

• EX`

  • $L(x,y,z)=(x+z,y)$ find the range(L) and Ker(L)

$$\begin{array}{r}range(L)=(\left[\begin{array}{c}x+z\\ y\end{array}\right]|x,y,z\in \mathbb{R})=(\left[\begin{array}{c}s\\ t\end{array}\right]|s,t\in \mathbb{R})\\ ={\mathbb{R}}^2\\ ker(L)=(\left[\begin{array}{c}x\\ y\\ z\end{array}\right]|\left[\begin{array}{c}x+z\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right])\\ ⟹x+z=0,y=0⟹z=t,x=-t\end{array}$$

Examples

• EX1

  • $T(x,y,z)=(x-y,y-z)$ is a linear mapping because:

$$\begin{array}{r}let\vec{a}=\left[\begin{array}{c}a1\\ a2\\ a3\end{array}\right],\vec{v}=\left[\begin{array}{c}v1\\ v2\\ v3\end{array}\right]\\ T(s\vec{a}+t\vec{v})=T(\left[\begin{array}{c}sa1+tv1\\ sa2+tv2\\ sa3+tv3\end{array}\right])\end{array}$$