Finding the Inverse of a Linear Transformation

[UrbanXrisis]

how would one find the inverse of the linear transformation:

y_1=4x_1-5x_2
y_2=-3x_1+4x_2

this was never taught in class, could someone give a little advice as how I would do this?

I know the answer has to be in the form of

x_1=ay_1+by_2
x_2=cy_1+dy_2

could someone explain this process?

 

[NateTG]

You're solving the system of equations for x_1 and x_2.


One way to do it would be to solve the first equation for x_1 and then substitute into the second equation.

Another method would be to add the equations together using suitable coefficitents so that one of the x's is eliminated, and then solve for the other.

In principle, this should be no different than dealing with, for example:
9=4x_1-5x_2
7=-3x_1+4x_2

 

[neutrino]

Yet another way is to write the equations in matrix form. (Left) Multiply both sides by the inverse of the coefficient matrix.