Interactive Linear Algebra
Dan Margalit & Joseph Rabinoff · Georgia Tech · Open textbook
Read §3.1 for matrix transformations and §3.4 for composition/matrix multiplication.
Open the sourceA matrix is an action
Its columns tell you where the basis arrows go.
Prerequisites: 06 · Coordinates are not the arrow, 07 · Independence: when a choice adds something
Understand
Objectives
- Predict a matrix output from its basis-image columns.
- Compute two compositions in their actual order.
- Reject a nonzero translation as a linear map.
Start with something you can see
On a square grid, shear every point (x,y) to (x+y,y). Predict where the two standard basis arrows land: (1,0) stays fixed and (0,1) becomes (1,1). Those two outputs determine the image of any linear combination. The creator video illustrates columns as basis images at 2:06. [1]
Give the idea a precise name
A function T:ℝ²→ℝ² is linear when T(u+v)=T(u)+T(v) and T(cu)=cT(u) for plane vectors u,v and real scalars c. Its matrix A records the images of the standard basis vectors as columns. Given linear maps A and B on ℝ², the product AB represents their composition with B applied first.
A(x,y)=x·column₁(A)+y·column₂(A)
A names the linear map and its matrix in standard coordinates; x,y are the input coefficients, and column₁ and column₂ are the images of e₁ and e₂. In ABv, first form Bv, then apply A. The jth column of AB is A applied to the jth column of B.
Work one small world
For A=[[1,1],[0,1]], the basis images are (1,0) and (1,1), so A(2,3)=2(1,0)+3(1,1)=(5,3). Its determinant 1 preserves oriented unit-square area. Let B=[[0,−1],[1,0]], a quarter-turn. For v=(1,1), Bv=(−1,1) and ABv=(0,1); Av=(2,1) and BAv=(−1,2). Order changes the result.
| Order on v=(1,1) | First step | Second step |
|---|---|---|
| ABv | Bv=(−1,1) | A(Bv)=(0,1) |
| BAv | Av=(2,1) | B(Av)=(−1,2) |
Counterexample to “every movement is a linear matrix action”: T(x,y)=(x+1,y) translates the origin to (1,0). Every linear map sends zero to zero, so this translation is affine, not linear. Likewise ABv and BAv above show that matrix order cannot generally be swapped.
THE BRIDGEAn invertible matrix may change coordinates and angles, but it preserves linear dependencies among vectors.
WHERE THIS IDEA STOPSThe columns describe a linear map in chosen input and output bases. A nonzero translation is affine, and multiplication need not commute.
Experiment
Use the existing matrix lab to inspect A as a shear; compute B as a quarter-turn on paper. Predict both compositions from the two-step table before evaluating them.
Why do two basis images determine the image of every vector in the plane?
Check & explain
For A=[[1,1],[0,1]], what is the first coordinate of A(3,4)?
For A and B in the worked example, which is AB(2,−1)?
- (3,2)
- (1,1)
- (1,−1)
Teach back
Read the columns of A and B as basis images, compute ABv and BAv for v=(1,1), and explain why a translation is not linear.
- Identify both images of the basis.
- Apply maps in right-to-left order.
- Check the image of zero for the translation.
Read deeper
Mathematics for Machine Learning
Deisenroth, Faisal & Ong · Freely readable textbook
Read §2.2 on matrices, printed p.22, and §2.7 on linear mappings, printed p.48.
Open the sourceThree-dimensional linear transformations | Chapter 5, Essence of linear algebra
3Blue1Brown · Video · 5 min
Watch the 5-minute creator video for columns as images of basis vectors.
Open the source30. Linear Transformations and Their Matrices
Gilbert Strang · MIT OpenCourseWare · Lecture excerpt · 0:00–20:00 of 49 min
Watch 0:00–20:00 of the 49-minute MIT lecture for linear transformations.
Open the source31. Change of Basis; Image Compression
Gilbert Strang · MIT OpenCourseWare · Lecture · 50 min
Full 50-minute change-of-basis lecture; distinguish a map from its matrix description.
Open the source