“Transformation” in the term “linear transformation” means function. That is, it takes an input and spits out an output. In the case of linear algebra, it will take in a vector and spit out another vector.
“Linear” in the same term means that when you apply the function, all straight lines, remain straight lines. None bend or curve. And that the origin remains fixed in place.
Basically, what is happening is that applying the function to the entire span moves each vector by the same amount, shifting and stretching (but not bending), the grid.
The grey grid above is the original, while the blue is the linearly transformed grid. See how the lines remain straight and the origin fixed.
The easy way to judge a grid to determine if the transformation was linear is to ask: “Are the grid lines parallel AND evenly spaced”.
Connections
Scaling Means To Multiply A Vector
Link Explanation: While scaling means to multiply a vector by a constant, stretching it out, a linear transformation often multiplies a vector, or the span, by another vector. This stretches the entire grid and warps the space.
Reference
https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3