Matrix multiplication as composed linear maps
Assistance allowed only after prediction and observed evidence are recorded.
Prerequisites
Complete Day 22; use the listed recovery link if matrix-vector products are shaky.
Learning Objective
Compute matrix products entry-by-entry as row-times-column dot products, interpret A*B as the composed map 'apply B then apply A', and demonstrate that matrix multiplication is not commutative.
Instructions
Entry (A*B)_ij is the dot product of row i of A with column j of B. Geometrically (A*B)*x = A*(B*x): the product matrix applies the inner map first, so composition reads right-to-left. Because composition depends on order, A*B and B*A generally differ.
Practice & Prediction Log
Let A = [[1,2],[3,4]] and B = [[0,1],[1,0]] (B swaps a vector's coordinates). Predict both products before computing: (a) A*B; (b) B*A. Are they equal?
For A*B, entry (1,1) is row 1 of A . column 1 of B = 1*0 + 2*1. Complete all four entries before starting the second product.
(a) A*B = [[1*0+2*1, 1*1+2*0], [3*0+4*1, 3*1+4*0]] = [[2,1],[4,3]]: B first swaps the coordinates, then A acts. (b) B*A = [[0*1+1*3, 0*2+1*4], [1*1+0*3, 1*2+0*4]] = [[3,4],[1,2]]: A acts first, then the swap. A*B != B*A, so matrix multiplication is not commutative; the order of transformations matters.
You compute both products correctly, explain the right-to-left composition reading, and give one geometric reason the two orders differ (swap-then-map vs map-then-swap).
Authoritative Source Resources & Provenance
Essence of Linear Algebra
Relevant Section / Range: Chapter 4: Matrix multiplication as composition
Verified Version & Date: 2016-2023 (Verified: 2026-10-30)
Unequaled geometric and visual intuition for vectors, linear transformations, matrix multiplication, and inverses before tensor operations.
High-school coordinate geometry
Geometric explanations, worked formulas, and coordinate exercises are fully provided as original lesson text.
Compared Candidate Sources & Selection Analysis (2)
Established textbook/lecture approach, strong mathematical rigor alongside 3Blue1Brown visual intuition.
Theoretical determinant-free proof structure is mathematically elegant but less aligned with computational tensor engineering.
18.06 Linear Algebra (Prof. Gilbert Strang)
Relevant Section / Range: Lecture 3: matrix multiplication and its four viewpoints
Verified Version & Date: Spring 2010 (Verified: 2026-10-30)
Established row-picture and column-picture treatment of linear systems, matrix products, and invertibility; rigorous complement to the 3Blue1Brown visual series.
High-school algebra and coordinate geometry
All row/column-picture derivations and worked examples are written as original lesson text and run offline.
Compared Candidate Sources & Selection Analysis (3)
Selected as co-primary: geometric visualizations paired with Strang's algebraic rigor.
Determinant-free theoretical treatment is elegant but further from the computational shape accounting required here.
Interactive figures are attractive, but the absence of a declared open license makes rights disposition unclear; kept out of the resource set.