Coordinates as a choice, change of basis

What you’ll learn

This is lesson 13 of Track 4 (Visual Math: Linear Algebra). The very first lesson flagged that a vector’s coordinates are “a description in a particular coordinate system, not the vector itself,” and promised to return to it; this is that lesson. By the end you will be able to translate a vector’s coordinates from one basis to another and back, using a basis matrix M and its inverse, and express a whole transformation in a different basis with the sandwich M^-1 · A · M. The real payoff is the mental shift: coordinates are a choice of language, not an absolute fact, and the same arrow or the same transformation simply reads differently in different bases. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.

Where this fits

This is lesson 13 of 15, the fourth lesson of Phase 3 (advanced perspectives). It follows Cramer’s rule in sequence but leans most on the inverses lesson (the M^-1 that makes translation reversible) and on the very first lesson’s flag about coordinates being relative. It also sets up the finale: once you know a transformation’s matrix depends on the basis, the natural question is which basis makes it simplest. The next lesson, Eigenvectors and eigenvalues, answers that, finding the basis in which a transformation becomes pure scaling.

Before you start

Prerequisites: the genuinely load-bearing background is Inverses, column space, and null space (you will use M^-1 and the det(M) ≠ 0 condition) and Linear transformations as moves (a matrix is where the basis lands, the column reading this lesson reuses). It also pays to recall the first lesson’s point that coordinates are relative. You should be comfortable inverting a 2x2 matrix with the shortcut formula. The practice is pen and paper.

About the math

The arithmetic is matrix-vector products and one 2x2 inverse, the most computation-heavy lesson in the track but with no new techniques. You will translate vectors between two bases both ways, confirm a round trip, and compute a transformation’s matrix in a new basis as M^-1 · A · M, verifying it against following a vector through the three steps. The concepts (coordinates as relative, the sandwich as three bundled steps) are the part worth slowing down for.

By the end, you’ll be able to

• Explain why a vector’s coordinates are relative to a chosen basis rather than absolute
• Build the basis matrix M (columns are the other basis in your coordinates) and use it to translate coordinates into your basis
• Use M^-1 to translate coordinates into the other basis, and verify a round trip
• Express a transformation in another basis with the sandwich M^-1 · A · M
• Connect change of basis to PCA, whitening, and other dimensionality-reduction techniques

Time and difficulty

• Read time: about 12 minutes
• Practice time: about 15 minutes (a translate-both-ways exercise, a transform-in-a-new-basis drill, and flashcards)
• Difficulty: standard, computation-heavy (the most arithmetic of any lesson in the track, though no new techniques; the depth is the coordinates-are-relative reframing)