How do I show that $T$ is invertible? I'm really stuck on these linear transformations, so I have $T(x_1,x_2)=(-5x_1+9x_2,4x_1-7x_2)$, and I need to show that $T$ is invertible. So would I pretty much just say that this is the matrix: $$\left[\begin{matrix}-5&9\\4&-7\end{matrix}\right]$$ Then it's inverse must be $\frac{1}{(-5)(-7)-(9)(4)}\left[\begin{matrix}-7&-9\\-4&-5\end{matrix}\right]=\left[\begin{matrix}7&9\\4&5\end{matrix}\right]$. But is that "showing" that $T$ is invertible? I'm also supposed to find a formula for $T^{-1}$. But that's the matrix I just found right?
 A: A fast way to check if a matrix is invertible, is to calculate $\det(T)$. If it's equal to $0$, you can't invert $T$, otherwise you can.
To find the general formula to invert a $2 \times 2$ matrix, try inverting one with $a,b,c,d$ as elements.
Spoiler alert : $$
\begin{bmatrix}
a & b \\
c & d 
\end{bmatrix}^{-1}= 
\frac{1}{\det(T)}
\begin{bmatrix}
d & -b \\
-c & a 
\end{bmatrix}
$$
A: Another way: Note that $T(1,0)=(-5,4)$ and $T(0,1)=(9,-7)$ are linearly independent vectors. Since $T$ maps a two-element basis to two linearly independent vectors, then it is invertible.
A: you can choose $\{(1,0) ,(0,1)\}$ as a base for your domain space then with attention to linear algebra matrix of $T$ will be $\begin{bmatrix}
-5 &9 \\ 
 4&-7 
\end{bmatrix}$and this a invertible , and any other matrix with other base for domain is ~ to this matrix
A: I think it would be more in the spirit of the question (it sounds like it is an exercise in a course or book) to write down a linear map $S$ such that $S\circ T$ and $T\circ S$ are both the identity - the matrix you have written down tells you how to do this. Then such an $S$ is $T^{-1}$.
You should also note that there are different matrices that can represent the map $T$, but it is true that checking that any such matrix is invertible amounts to a proof that $T$ is invertible.
This non-uniqueness of matrices also means that I would disagree that the matrix you found is the same thing as "a formula for $T^{-1}$". You should say what the map $T^{-1}$ does to a point $(y_1,y_2)$.
A: You can also show that $\ker T=\{\vec{0}\}$ by solving linear equations system.
