References: Matrices between dimensions

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 8: "Nonsquare matrices as transformations between dimensions"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/nonsquare-matrices
  Series index: https://www.3blue1brown.com/?topic=linear-algebra
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• Nonsquare matrices as transformations between dimensions (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Seeing a 2D grid lift off the page and settle as a tilted plane in 3D (the embedding), and seeing 3D space flatten down onto a plane (the projection), makes the direction of a rectangular mapping immediate. Short and visually clear.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The previous chapter (inverses, column space, null space) supplied the rank and null-space tools this lesson reuses; the next (Dot products and duality) looks at the extreme case of a one-row matrix that turns a vector into a single number.

• Khan Academy: Linear algebra for a slower, exercise-driven treatment of matrix transformations, rank, and dimension, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• Inverses, column space, and null space (previous lesson). This lesson reuses rank, column space, and null space without change; the only new thing is that input and output dimensions can now differ, which makes the rank-nullity balance more interesting (a full-rank projection still has a null space).

• Dot products and duality (next lesson). A 1-by-n matrix is the most extreme rectangular case: it takes an n-dimensional vector and outputs a single number. That turns out to be exactly the dot product, and the next lesson shows why a vector and a “vector-to-number” transformation are two views of the same object.