I need to determine a matrix that can be used to:
a) transform $x$ and $y$ using equations
$$ \left\{ \begin{array}{c} x'=3x + 4y \\ y'=-x + 2y \\ \end{array} \right. $$
Then I need to:
b) Transform a triangle $(0,0),(1,0),(1,1)$ using this transformation
c) What are the coordinates of point $(10,10)$ in the basis of transformation?
Could I have some advise? Thanks in advance.
$\left\{ \begin{array}{c} x'=3x + 4y \\ y'=-x + 2y \\ \end{array} \right. \Rightarrow \left\{ \begin{array}{c} x'=(3,4)(x,y)\\ y'=(-1,2)(x,y) \\ \end{array} \right.$
$\left\{ \begin{array}{c} x'=(3,4)(x,y)\\ y'=(-1,2)(x,y) \\ \end{array} \right. \Rightarrow \begin{pmatrix} 3 & 4 \\ -1 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} x` \\ y` \end{pmatrix}$
$\begin{pmatrix} 3 & 4 \\ -1 & 2 \end{pmatrix}\begin{pmatrix} 0 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \end{pmatrix}$
$\begin{pmatrix} 3 & 4 \\ -1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 3 \\ -1 \end{pmatrix}$
$\begin{pmatrix} 3 & 4 \\ -1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} 7 \\ 1 \end{pmatrix}$
as itcan be seen, image of triangle is also a triangle (because kernel of matrix is $\{0\}$)
