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Linear Algebra

Matrices Are Transformations

Start here: once a matrix is a grid transform, multiplication, determinants, and inverses connect naturally.

The root idea

A matrix is a grid transformer

Change the numbers and watch where the i/j axes and the 1×1 square go.

Choose a matrix mood
A
The left column is where i lands; the right column is where j lands.
i lands at(2, 0)
j lands at(1, 1)
Area scale2 x

The grid gets pulled right; columns can tilt the axes.

Grid after A transforms it
det = 2

Light lines are the old grid; blue lines are the grid after matrix A.

From this picture:Multiply = compose transformationsDet = area scaleInverse = can we undo?
Speed 1.0x
Credits & References