I am looking for an explanation of (in)homogeneity of linear systems of equations from a perspective of linear transformations.
For example, the following connections between linear transformations and linear systems of equations I understand:
1) If the system is homogenous it has at least $\mathbf{0}$ as a solution.
Transformation perspective:
$\mathbf{0}$ is a solution, as $T\mathbf{0} = \mathbf{0}$ for all linear transformations $T$; trivial from definition of matrix multiplication. There might be other vectors that $T$ sends to $0$.
2) If the system is invertible, is has exactly one solution.
Transformation perspective:
If it is invertible, it is a bijective mapping, thus there is exactly one vector that satisfies the system. If there were more, it would not be bijective.
So, specifically I would like to understand from a transformations perspective the following:
From a linear transformations perspective, what does it mean for a system of equatiosn to be homogenous (apart from $\mathbf{0}$ being a trivial member of the nullspace of the transformation.)
From a linear transformations perspective, what does it mean for a system of equatiosn to be inhomogeneous
what are free variables from the perspective of linear transformations
from the perspective of linear transformations why are there infinitely many solutions if there are more variables than equations ( = basis vectors from the point of view of transformation?)