Differences between a matrix that represents a linear transformation and a matrix that represents a system of linear equations?

Question

I am pondering two facts:

1. Matrices represent a linear transformation (given a choice of basis)
2. Matrices can store the coefficients for a system of linear equations

Is there something deeper going on here? What's the connection between coefficients of a linear system of equations and coordinates for a linear transformation?

Answer 1

Here's the connection.

Let $M$ be the matrix in question and let $T$ be the linear transformation associated to $M$.

Now let's examine the given system of linear equations with coefficient matrix $M$. If you encode the variables in that system as a single column matrix $\vec x$, and if you encode the constants on the right hand side of that system as a single column matrix $\vec b$, then the entire system of equations can be rewritten as a single equation $T \vec x = \vec b$.

That can sometimes make it very straightforward, in a conceptual sense, to solve the original system of equations itself. For example in $M$ is a square matrix, and if $\det M \ne 0$, equivalently the matrix $M$ is invertible, equivalently the linear transformation $T$ is invertible, then the solution of the system is $$\vec x = T^{-1} \vec b $$