# What is the relationship between linear transformations and solutions to linear equations?

I think I am having trouble seeing the forest from the tree but I know that a linear transformation obeys L(v + w) = L(v) + L(w) where v,w are vectors and L(cv) = cL(v) where c is a scalar. Good. Geometrically they are those transformations that keep the grid lines parallel and evenly spaced and which keep the origin fixed.

I also know that lines of systems of equations are the same line if they share all solutions. Let us say there is a solution to a system of equations but only one and the lines geometrically will cross at some point. You can always set up a matrix with such a system call it matrix A.

What is the relationship between matrix A of a coordinate transformation and linearity? Matrix A still does something to the original coordinate frame does it not? The book explains how different equations of the lines intersect each other or not depending on solutions AND then in another chapter show linear transformations change the grid of a coordinate axis. But what's the relationship? Perhaps the fact that it is a line makes it "linear" but then they should not cross each other in order to keep the grid lines even should they? Thank you .