A linear map is geometry with discipline. It can stretch,
shear, or rotate space, but two promises never break: straight
lines stay straight, and the origin never moves.
Start with the standard grid and the two basis
arrows, e one and e two. They are the
coordinate rulers: every vector is built from these directions.
Now add the unit square. Right now its area is
one, and its edges align with the basis. This square
will show us exactly what the transformation does.
Apply the matrix A. The first basis arrow lands at the first column,
and the second arrow lands at the second column. Columns are the map.
Watch the full grid move at once. Cells tilt, squares
become parallelograms, and distances change unevenly. But every row and
column remains a family of lines - classic shear behavior.
Notice what survives: this diagonal was a line before, and
it is still a line after. And the origin, where
both basis arrows start, is still pinned in place.
The unit square became a larger parallelogram. Its area scale is
the determinant. Here det A is one point four, so every
tiny area patch gets multiplied by one point four.
Now lift the idea to three dimensions. A unit cube is
just three basis directions combined. The same linear rules still apply
in three D: planes stay planes, and the origin stays fixed.
When this transformed cube appears in graphics, robotics, or
data embeddings, it is the same story: new basis
vectors first, then every point follows by linear combination.
That is a linear map: choose where the basis goes,
and everything else is forced. Structure first, motion second.