Matrices as transformations

After this lesson you can: See a matrix as something that moves space, not a grid of numbers.

First: Vectors, and what they actually represent

Watch

• Linear transformations and matrices | Chapter 3, Essence of linear algebra - 3Blue1Brown, 11 min. Directly targets the lesson’s core shift: matrices as geometric transformations rather than number grids.
• Matrix multiplication as composition | Chapter 4, Essence of linear algebra - 3Blue1Brown, 10 min. Extends the transformation picture to matrix multiplication, correcting rote multiplication with a visual meaning.
• 30. Linear Transformations and Their Matrices - MIT OpenCourseWare, 49 min. A fuller university treatment for learners who want a slower, more formal consolidation of the idea.

Notes

A matrix is best understood as a rule for moving every vector in space. In two dimensions, it takes the whole coordinate plane and stretches, squashes, rotates, shears, reflects, or collapses it. The numbers in the matrix are not the main story; they are a compact way to describe where the transformation sends the basic direction vectors.

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