4.4.2 Change of Coordinates

What is a change of coordinates

• Let $B=({\vec{v}}_1,\dots ,{\vec{v}}_n)$ and $c=({\vec{w}}_1,\dots ,{\vec{w}}_n)$ be bases of ${\mathbb{R}}^n$. the change of coordinates matrix from B-coordinates to C-coodinates is the matrix $P_{B\to C}=[[{\vec{v}}_1][{\vec{v}}_2]]$

• basically we change the basis of a vector so the coordinates become a different coordinate system

• In general, conversion from one coordinate system to another is a 3.2 linear mappings

Ex

• Let $S$ denote the standard basis ${\vec{e}}_1,{\vec{e}}_2,{\vec{e}}_3$ and $\vec{x}=(1,2,3)$ what is $(\vec{x}{)}_S$

$$\begin{array}{r}(1,2,3)=1(1,0,0)+2(0,1,0)+3(0,0,1)=1{\vec{e}}_1+2{\vec{e}}_2+3{\vec{e}}_3\\ (\vec{x}{)}_S=\vec{x}=(1,2,3)\end{array}$$

• example ${\mathbb{R}}^2$ let $B=(\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ -1\end{array}\right])$ and $\vec{x}=(3,1).$ what is $(\vec{x}{)}_B$

  • We need to write $\vec{x}=c_1{\vec{v}}_1+c_2{\vec{v}}_2$

$$\begin{array}{r}{c}_1\left[\begin{array}{c}1\\ 1\end{array}\right]+c_2\left[\begin{array}{c}1\\ -1\end{array}\right]=\left[\begin{array}{c}3\\ 1\end{array}\right]\\ RREF(\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\end{array}\right]),c_1=2,c_2=1\\ (\vec{x}{)}_B=(2,1)\end{array}$$

• Same B as above. This time, we give ${\vec{x}}_B=(-1,1)$ what is in standard coordinates

$$\begin{array}{r}\vec{x}=c_1{\vec{v}}_1+c_2{\vec{v}}_2\\ -1\left[\begin{array}{c}1\\ 1\end{array}\right]+1\left[\begin{array}{c}1\\ -1\end{array}\right]=\left[\begin{array}{c}0\\ -2\end{array}\right]\end{array}$$