Let $L: \mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear operator such that $L((1,2)^T)=(-2,3)^T$ and $L((1,-1)^T)=(5,2)^T$, then find $L((7,5)^T)$.

Question

Let $L: \mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear operator. If $L((1,2)^T)=(-2,3)^T$ and $L((1,-1)^T)=(5,2)^T$, find the value of $L((7,5)^T).$

Is there a way to solve these kinds of problems? I only know if $L(\alpha v_1+\beta v_2)=\alpha L(v_1)+\beta L(v_2)$, then the vector space is said to be a linear transformation.

Answer 1

Let your transformation matrix $L=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Then $L((1,2)^T)=(-2,3)^T$ and $L((1,-1)^T)=(5,2)^T$ give $$\begin{align}&a+2b=-2 &c+2d=3\\&a-b=5 &c-b=2.\end{align}$$ Solving we get $L=\begin{bmatrix}\dfrac{8}{3}&\dfrac{-7}{3}\\\dfrac{7}{3}&\dfrac{1}{3}\end{bmatrix}$. $\:$So $L((7,5)^T)=\begin{bmatrix}7\\18\end{bmatrix}$.

Answer 2

Right, so you have to find $\alpha$ and $\beta$ with $$ \alpha(1,2)+\beta(1,-1)=(7,5)$$