Suppose that \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is a linear transformation satisfying \(T(1,1) = (5, 3)\) and \(T(2,3) = (7, 9)\). Find the standard matrix of \(T\).
Solution
1. Let\[ u=T(1,0),\qquad v=T(0,1). \]Then\[ u+v=(5,3), \qquad 2u+3v=(7,9). \]
This follows from linearity and the decompositions \((1,1)=(1,0)+(0,1)\) and \((2,3)=2(1,0)+3(0,1)\).
2. We have\[ v=(-3,3). \]
Subtract twice the first equation in 1 from the second: \[ (2u+3v)-2(u+v)=v=(7,9)-2(5,3)=(-3,3). \]
3. We have\[ u=(8,0). \]
From \(u+v=(5,3)\) and 2, \[ u=(5,3)-(-3,3)=(8,0). \]
4. Therefore the standard matrix of \(T\) is\[ [T]=\begin{bmatrix}8&-3\\0&3\end{bmatrix}. \]
The columns of the standard matrix are \(T(1,0)=u\) and \(T(0,1)=v\).