A linear transformation moves every vector by moving the basis directions first, then extending that action linearly to the whole space. This chapter begins with 2D grid deformation, develops standard examples such as scaling, shear, rotation and reflection, then connects them to matrix-vector multiplication and one-to-one and onto conditions.
A matrix is best understood as a rule that sends basis vectors to new directions and then sends every other vector by the same linear recipe.
The chapter ends by comparing maps from ℝ² to ℝ³ and from ℝ³ to ℝ², showing how rank controls embedding, collapse and coverage of the codomain.
Because every vector is built from the basis vectors. Once we know where the basis vectors go, linearity forces the image of every combination. That is why the columns of M encode the full action of the transformation.
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Because two independent input directions can remain independent after the map, so no dimension collapses. But two directions are still not enough to fill a three-dimensional codomain. The image becomes a plane in ℝ³, not all of ℝ³.