# 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 $L((7,5)^T)$.

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.

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 $$L((7,5)^T)=\begin{bmatrix}7\\18\end{bmatrix}$$.
• @ShadowZ: If we have a linear transformation from $R^n$ ton $R^m$, the corresponding matrix associated with the transformation will have dimension $m\times n$ – Yadati Kiran Nov 18 '18 at 10:37
Right, so you have to find $$\alpha$$ and $$\beta$$ with $$\alpha(1,2)+\beta(1,-1)=(7,5)$$