Computation Lab: Matrix Composition and Basis Action

An interactive lab for seeing how a matrix acts on basis vectors and how composing two maps differs from applying either one alone.

Keywords

computation, simulation, visualization, matrices, composition

1 Lab Goal

This lab helps you see one specific fact:

a matrix is fully determined by what it does to the basis vectors, and composing two maps creates a new matrix whose action depends on order.

2 Math Question

How do shear and scale maps affect:

• the images of \(e_1\) and \(e_2\)
• the image of a sample vector
• the composed map \(AB\) versus \(BA\)

3 Model or Setup

We use two linear maps in \(\mathbb{R}^2\):

\[ B = \begin{bmatrix} 1 & s \\ 0 & 1 \end{bmatrix}, \qquad A = \begin{bmatrix} \lambda & 0 \\ 0 & 1 \end{bmatrix}. \]

Here \(B\) is a shear and \(A\) rescales the first coordinate.

4 Parameters and Controls

• Shear s
• Horizontal scale λ
• Sample vector x

Defaults are 0.8, 1.5, and \(x=(1,1)\).

5 Code and Simulation

viewof shear = Inputs.range([-2, 2], {
  value: 0.8,
  step: 0.05,
  label: "Shear s"
})

viewof scaleX = Inputs.range([0.25, 2.5], {
  value: 1.5,
  step: 0.05,
  label: "Horizontal scale λ"
})

viewof x1 = Inputs.range([-2, 2], {
  value: 1,
  step: 0.05,
  label: "Sample vector x-coordinate"
})

viewof x2 = Inputs.range([-2, 2], {
  value: 1,
  step: 0.05,
  label: "Sample vector y-coordinate"
})

B = [[1, shear], [0, 1]]
A = [[scaleX, 0], [0, 1]]

applyMat = (M, v) => ({
  x: M[0][0] * v.x + M[0][1] * v.y,
  y: M[1][0] * v.x + M[1][1] * v.y
})

matMul = (M, N) => [
  [
    M[0][0] * N[0][0] + M[0][1] * N[1][0],
    M[0][0] * N[0][1] + M[0][1] * N[1][1]
  ],
  [
    M[1][0] * N[0][0] + M[1][1] * N[1][0],
    M[1][0] * N[0][1] + M[1][1] * N[1][1]
  ]
]

e1 = ({x: 1, y: 0})
e2 = ({x: 0, y: 1})
x = ({x: x1, y: x2})

AB = matMul(A, B)
BA = matMul(B, A)

images = [
  {label: "e1", ...e1},
  {label: "e2", ...e2},
  {label: "x", ...x},
  {label: "ABx", ...applyMat(AB, x)},
  {label: "BAx", ...applyMat(BA, x)}
]

segments = [
  {x1: 0, y1: 0, x2: e1.x, y2: e1.y, kind: "e1"},
  {x1: 0, y1: 0, x2: e2.x, y2: e2.y, kind: "e2"},
  {x1: 0, y1: 0, x2: x.x, y2: x.y, kind: "x"},
  {x1: 0, y1: 0, x2: applyMat(AB, x).x, y2: applyMat(AB, x).y, kind: "ABx"},
  {x1: 0, y1: 0, x2: applyMat(BA, x).x, y2: applyMat(BA, x).y, kind: "BAx"}
]

Plot.plot({
  height: 430,
  x: {label: "x", domain: [-4, 4]},
  y: {label: "y", domain: [-4, 4]},
  color: {legend: true},
  marks: [
    Plot.gridX({strokeOpacity: 0.15}),
    Plot.gridY({strokeOpacity: 0.15}),
    Plot.ruleX([0], {strokeOpacity: 0.25}),
    Plot.ruleY([0], {strokeOpacity: 0.25}),
    Plot.link(segments, {
      x1: "x1",
      y1: "y1",
      x2: "x2",
      y2: "y2",
      stroke: "kind",
      strokeWidth: 3,
      markerEnd: "arrow"
    }),
    Plot.dot(images, {x: "x", y: "y", fill: "label", r: 5, tip: true})
  ]
})

Inputs.table([
  {map: "A", row1: `[${A[0][0].toFixed(2)}, ${A[0][1].toFixed(2)}]`, row2: `[${A[1][0].toFixed(2)}, ${A[1][1].toFixed(2)}]`},
  {map: "B", row1: `[${B[0][0].toFixed(2)}, ${B[0][1].toFixed(2)}]`, row2: `[${B[1][0].toFixed(2)}, ${B[1][1].toFixed(2)}]`},
  {map: "AB", row1: `[${AB[0][0].toFixed(2)}, ${AB[0][1].toFixed(2)}]`, row2: `[${AB[1][0].toFixed(2)}, ${AB[1][1].toFixed(2)}]`},
  {map: "BA", row1: `[${BA[0][0].toFixed(2)}, ${BA[0][1].toFixed(2)}]`, row2: `[${BA[1][0].toFixed(2)}, ${BA[1][1].toFixed(2)}]`}
])

6 What To Observe

• The images of \(e_1\) and \(e_2\) are exactly the columns of each matrix.
• The composed images \(ABx\) and \(BAx\) usually differ.
• Changing the shear alters how much the second basis direction spills into the first coordinate.
• Changing the scale changes the effect of composition depending on when the scaling happens.

7 Interpretation

The plot is showing three core claims from the concept page:

• columns record basis images
• one matrix is one linear rule
• composition becomes matrix multiplication, and order matters

8 Failure Modes and Numerical Cautions

• This lab is deterministic and tiny, so it hides numerical conditioning issues.
• In higher dimensions you cannot inspect basis images visually as easily, but the algebra stays the same.
• A composition can look visually similar for one sample vector even when the matrices differ strongly.

9 Reproducibility Notes

• execution engine: Observable JS
• randomness: none
• libraries: Quarto OJS with Plot and Inputs
• render mode: interactive client-side

10 Extensions

• replace the scaling map by a rotation-like matrix
• compare one map written in two different bases

11 Sources and Further Reading

• MIT 18.06SC: Linear Transformations and their Matrices - First pass - official source for the basis-image and composition viewpoint. Checked 2026-04-24.
• Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares - Second pass - good computational reinforcement on transformations and maps. Checked 2026-04-24.
• Deep learning, transformers and graph neural networks: a linear algebra perspective - Paper bridge - useful later, when you want to see modern learned operators through the same lens. Checked 2026-04-24.