Matrix multiplication as composition

What you’ll learn

This is lesson 4 of Track 4 (Visual Math: Linear Algebra). The previous lesson showed that one matrix is one move; this lesson answers the obvious follow-up, what records two moves in a row. By the end you will be able to take two transformations, compose them, and verify that the result matches the matrix product, computed the meaning-preserving way (apply the left matrix to each column of the right one). You will see why the product reads right to left like nested functions, why AB is generally not BA (order is the sequence of physical moves), and why grouping never matters even though order always does. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.

Where this fits

This is lesson 4 of 15, the fourth step of Phase 1 (geometric foundations). It chains the single transformations from Linear transformations as moves: computing a product is just that lesson’s matrix-vector product, run once per column. The next lesson, Transformations in 3D, keeps every rule identical (a matrix is where the basis lands, multiplication is composition) and adds a third basis vector, so a transformation becomes a 3-by-3 matrix.

Before you start

Prerequisites: the previous lesson, Linear transformations as moves. You need to be comfortable reading a matrix as the two transformed basis vectors written as columns, and computing M · v as a linear combination of those columns, because the whole computation here is that operation applied once to each column. The practice is pen and paper.

About the math

The arithmetic is the same scale-and-add you have been doing, now run a couple of extra times. To multiply two 2x2 matrices, you apply one matrix to each column of the other, which is two matrix-vector products. The practice walks two products by hand and checks non-commutativity by computing both orders. No new formulas, and the emphasis stays on the meaning (composition) rather than the rote recipe.

By the end, you’ll be able to

• Explain matrix multiplication as composition of transformations, where AB means “first apply B, then apply A”
• Read a matrix product right to left, and justify it from (AB)·v = A·(B·v)
• Compute a 2x2 product column by column (apply the left matrix to each column of the right matrix)
• Show by example that matrix multiplication is not commutative (AB is generally not BA)
• State that matrix multiplication is associative ((AB)C = A(BC)) and explain why grouping does not change the result

Time and difficulty

• Read time: about 10 minutes
• Practice time: about 15 minutes (a compose-two-transformations exercise, a predict-then-verify drill, and flashcards)
• Difficulty: intro (foundational; the arithmetic is the previous lesson’s matrix-vector product run once per column)