Question

A transformação linear T:R^2 pertence R^2 onde T(1,2)=(6,-4) e t(2,1)=(9,1) é melhor representada pela alternativa;
A)T((X,Y))=4X+Y,2X+3Y)
B)T((X,Y))=(3X+7Y+Y)
C)T((X,Y))=(2X+5Y,X-2Y)
D)T((X,Y))=(X+5Y,3X-Y)

Answer 1

$T(1,2)=(6,-4)\\ T(2,1)=(9,1)$

Sean $e_1=(1,0)$ e $e_2=(0,1)$ las bases canónicas de $\mathbb R^2$, entonces

$T(1,2)=T(e_1+2e_2)=T(e_1)+2T(e_2)=(6,-4)\\ T(2,1)=T(2e_1+e_2)=2T(e_1)+T(e_2)=(9,1)\\ \\ \begin{cases} T(e_1)+2T(e_2)=(6,-4)\\ 2T(e_1)+T(e_2)=(9,1) \end{cases}$

$\begin{cases} T(e_1)+2T(e_2)=(6,-4)\\ -4T(e_1)-2T(e_2)=(-18,-2) \end{cases}\\ \\ \\ -3T(e_1)=(-12,-6)\to \boxed{T(e_1)=(4,2)\wedge T(e_2)=(1,-3)}\\ \\ T(x,y)=xT(e_1)+yT(e_2)\\ \\ T(x,y)=x(4,2)+y(1,-3)\\ \\ \\ \boxed{\boxed{T(x,y)=(4x+y,2x-3y)}}$