Linear transformation
A function on vectors that keeps addition and scaling intact. Every such function in coordinates is multiplication by a matrix.
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Analogy
A funhouse mirror that stretches and turns every point but keeps straight lines straight.
Example
Matrix [[0,-1],[1,0]].(1,0) goes to (0,1).(0,1) goes to (-1,0).(2,3) goes to (-3,2).
In practice
Rotations, reflections and scalings in graphics and robotics are all linear transformations.
Questions people ask
How do I find the matrix of a linear transformation?
See where the transformation sends each basis vector, and use those results as the columns. A 90 degree rotation sends (1,0) to (0,1) and (0,1) to (-1,0), so the matrix columns are (0,1) and (-1,0). Linearity then handles every other vector automatically, since each is a combination of the basis vectors.
Why are translations not linear transformations?
A linear transformation must send the zero vector to itself. Moving every point by adding (1,2) shifts the origin, so it breaks the rule. Graphics software handles translations with a trick using one extra coordinate, so it can still use matrices. Rotation, scaling and reflection do keep the origin fixed, so they are linear.