References: Change of basis

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 13: "Change of basis"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/change-of-basis
  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

• Change of basis (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the same arrow described by two grids at once, and watching the M^-1 · A · M sandwich translate a transformation between those grids, makes “coordinates are a choice” click in a way the algebra alone does not. Grant’s Jennifer-and-us framing is the one this lesson borrows. About thirteen minutes.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The inverses chapter supplied the M^-1 that makes the round trip work; the next (Eigenvectors and eigenvalues) finds the special basis in which a transformation becomes pure scaling.

• Khan Academy: Linear algebra for a slower, exercise-driven treatment of change of basis and coordinate systems, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• What a vector actually is (the first lesson). That lesson flagged the exact point this one develops: coordinates are a description of the vector in a chosen frame, not the vector itself. This lesson is the operational follow-through, twelve lessons later.

• Eigenvectors and eigenvalues (next lesson). Change of basis raises the question of which basis is best for a given transformation. Eigenvectors are the answer: in the eigenvector basis, a transformation becomes a diagonal matrix of pure stretch factors, the simplest possible description. The next lesson finds that basis.