Computation Lab: Linear Combinations and Span Geometry

An interactive lab for seeing how coefficients, generator directions, and dependence determine what vectors you can build.

Keywords

computation, simulation, visualization, span, vectors

1 Lab Goal

This lab helps you see one specific fact:

changing coefficients moves you inside the span you already have, but changing the generator directions changes the span itself.

2 Math Question

How do the coefficients and the angle between two generator vectors affect:

• the constructed vector
• whether the generators are independent
• whether the span is a line or the whole plane

3 Model or Setup

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

\[ v_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad v_2 = \begin{bmatrix} \cos \theta \\ \sin \theta \end{bmatrix}. \]

The constructed vector is

\[ x = a v_1 + b v_2. \]

4 Parameters and Controls

• Angle: controls whether \(v_2\) is nearly parallel to \(v_1\) or clearly independent
• Coefficient a: weight on \(v_1\)
• Coefficient b: weight on \(v_2\)

Default values are 50 degrees, 1.2, and 0.8.

5 Code and Simulation

viewof angleDeg = Inputs.range([0, 180], {
  value: 50,
  step: 1,
  label: "Angle of second vector (degrees)"
})

viewof coeffA = Inputs.range([-2, 2], {
  value: 1.2,
  step: 0.05,
  label: "Coefficient a"
})

viewof coeffB = Inputs.range([-2, 2], {
  value: 0.8,
  step: 0.05,
  label: "Coefficient b"
})

theta = angleDeg * Math.PI / 180
v1 = ({x: 1, y: 0})
v2 = ({x: Math.cos(theta), y: Math.sin(theta)})
target = ({
  x: coeffA * v1.x + coeffB * v2.x,
  y: coeffA * v1.y + coeffB * v2.y
})

detValue = v1.x * v2.y - v1.y * v2.x
independenceText =
  Math.abs(detValue) < 1e-6
    ? "Dependent generators: the span collapses to one line."
    : "Independent generators: the span is all of R^2."

segments = [
  {x1: 0, y1: 0, x2: v1.x, y2: v1.y, kind: "v1"},
  {x1: 0, y1: 0, x2: v2.x, y2: v2.y, kind: "v2"},
  {x1: 0, y1: 0, x2: target.x, y2: target.y, kind: "target"}
]

linePoints = Array.from({length: 121}, (_, i) => {
  const t = -3 + i * 0.05
  return {x: t * v1.x, y: t * v1.y}
})

Plot.plot({
  height: 430,
  x: {label: "x", domain: [-3, 3]},
  y: {label: "y", domain: [-3, 3]},
  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.line(linePoints, {x: "x", y: "y", stroke: "#cbd5e1", strokeWidth: 2}),
    Plot.link(segments, {
      x1: "x1",
      y1: "y1",
      x2: "x2",
      y2: "y2",
      stroke: "kind",
      strokeWidth: 3,
      markerEnd: "arrow"
    }),
    Plot.dot([{x: target.x, y: target.y}], {x: "x", y: "y", r: 6, fill: "#dc2626"})
  ]
})

md`
- Constructed vector: **(${target.x.toFixed(3)}, ${target.y.toFixed(3)})**
- Determinant of [v1 v2]: **${detValue.toFixed(4)}**
- Interpretation: **${independenceText}**
`

6 What To Observe

• As long as the angle is not 0 or 180 degrees, the two directions are independent.
• When the angle approaches 0 or 180, the determinant approaches zero and the span collapses toward a single line.
• The sliders for \(a\) and \(b\) move the red vector inside the span you already have; they do not create a new span.
• Dependence is not about one coefficient choice. It is about the geometry of the generators themselves.

7 Interpretation

This is the geometry behind the phrase

span is the set of all linear combinations.

When the generators are independent in \(\mathbb{R}^2\), their span is the whole plane. When they are dependent, all combinations stay on one line. The determinant here is a quick 2D witness of that change.

8 Failure Modes and Numerical Cautions

• Near-zero determinants can look nonzero numerically because of floating-point rounding.
• This is only a 2D picture. In higher dimensions, span and dependence still behave the same way, but the geometry is harder to draw.
• A visually small angle does not by itself prove exact dependence; exact dependence is an algebraic statement.

9 Reproducibility Notes

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

10 Extensions

• replace \(v_1\) by a non-axis-aligned vector
• add a third vector and test when it changes the span and when it is redundant

11 Sources and Further Reading

• MIT 18.06SC: Basis and Dimension - First pass - good official anchor for span and dependence. Checked 2026-04-24.
• Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares - Second pass - useful for computational and geometric phrasing. Checked 2026-04-24.
• Attention is All you Need - Paper bridge - later reminder that weighted vector mixtures remain central in modern ML systems. Checked 2026-04-24.