Question

Consider the linear transformations :

f:R^2→R^2 g:R^2→R^2

f (x,y)→(2x-y,3x+y) g(x,y)→(4x-2y,2x+y)

a) Determine the matrix of linear transformation f relative to the basis

{e1=(1,2),e2=(-1,1)}

b)dertimine (fog)(xy)

c) find g^-1

Answer 1

$f(x,y)=(2x-y,\ 3x+y),\ \ \ g(x,y)=(4x-2y, \ 2x+y)$

a)Let $A$ be the matrix of $f$ with respect to the standard basis,

$B$ be the matrix of $f$ with respect to the basis $e_1=(1,2), \ e_2=(-1,1)$ , and $P$ be the transition matrix.

$A=\begin{pmatrix} 2 & -1 \\ 3 & 1 \end{pmatrix}$

$P=\begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix}$ , $P^{-1}=\frac{1}{3} \begin{pmatrix} 1 & 1 \\ -2 & 1 \end{pmatrix}=\begin{pmatrix} 1/3 & 1/3 \\ -2/3 & 1/3 \end{pmatrix}$

$B=P^{-1}AP =\begin{pmatrix} 1/3 & 1/3 \\ -2/3 & 1/3 \end{pmatrix} \begin{pmatrix} 2 & -1 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix} =\begin{pmatrix} 1/3 & 1/3 \\ -2/3 & 1/3 \end{pmatrix} \begin{pmatrix} 0 & -3 \\ 5 & -2 \end{pmatrix}=\begin{pmatrix} 5/3 & -5/3 \\ 5/3 & 4/3 \end{pmatrix}$

b) $(f\circ g)(x,y) = f(4x-2y,\ 2x+y)=\big(2(4x-2y)-(2x+y),\ 3(4x-2y)+(2x+y)\big)= (6x-5y, 14x-5y)$

c) $g(x,y)=(4x-2y, \ 2x+y)=(z,w)$

Let $G$ be the matrix of $f$ : $G=\begin{pmatrix} 4 & -2 \\ 2 & 1 \end{pmatrix}$

Then $G^{-1}=\frac {1}{8} \begin{pmatrix} 1 & 2 \\ -2 & 4 \end{pmatrix}=\begin{pmatrix} 1/8 & 1/4 \\ -1/4 & 1/2 \end{pmatrix}$

$g^{-1}(z,w)=(\frac{z}{8}+\frac{w}{4}, \ -\frac{z}{4}+\frac{w}{2})$

Answer:

a) $B= \begin{pmatrix} 5/3 & -5/3 \\ 5/3 & 4/3 \end{pmatrix}$

b) $(f\circ g)(x,y) = (6x-5y, 14x-5y)$

c) $g^{-1}(z,w)=(\frac{z}{8}+\frac{w}{4}, \ -\frac{z}{4}+\frac{w}{2})$