Matrix Operations

Matrices aren’t just grids of numbers—they’re powerful tools for performing linear transformations. Stretching, rotating, reflecting, and projecting: matrix operations reshape space and reveal structure in data, geometry, and beyond. 🔄

Linear Transformation

A transformation (function or mapping) \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) denoted \(T\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a rule that assigns to each vector \(\mathbf{x} \in \mathbb{R}^n\) a vector \(T(\mathbf{x}) \in \mathbb{R}^m\).

Matrices serve as linear transformations

\[ T(\mathbf{x}) = A\mathbf{x} \]

The set \(\mathbb{R}^n\) is called the domain of \(T\), and the set \(\mathbb{R}^m\) is called the codomain of \(T\).

Assume a matrix \(A\) with dimensions \(m\) x \(n\), we can compute the dot product or matrix multiplication by

\[ A\mathbf{x} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \end{bmatrix} = \mathbf{b} \]

\[ A\mathbf{x} = \mathbf{b} \]

To perform the matrix multiplication \(A\mathbf{x}\), the number of columns in \(A\) must match the number of entries in the vector \(\mathbf{x}\).
That is, if \(A\) is an \(m \times n\) matrix and \(\mathbf{x}\) is an \(n \times 1\) column vector, the multiplication is valid and the result will be an \(m \times 1\) column vector.
If the dimensions do not align, the dot product is undefined.

viewof a11 = Inputs.range([-3, 3], {
  step: 0.1, 
  value: 1.5, 
  label: tex`a_{11}`, 
  width: 200
})

viewof a12 = Inputs.range([-3, 3], {
  step: 0.1, 
  value: 0.5, 
  label: tex`a_{12}`, 
  width: 200
})

viewof a21 = Inputs.range([-3, 3], {
  step: 0.1, 
  value: -0.5, 
  label: tex`a_{21}`, 
  width: 200
})

viewof a22 = Inputs.range([-3, 3], {
  step: 0.1, 
  value: 1.2, 
  label: tex`a_{22}`, 
  width: 200
})

// Vector x components
viewof x1 = Inputs.range([-5, 5], {
  step: 0.1, 
  value: 3, 
  label: tex`x_1`, 
  width: 200
})

viewof x2 = Inputs.range([-5, 5], {
  step: 0.1, 
  value: 2, 
  label: tex`x_2`, 
  width: 200
})

// Calculate transformed vector b = Ax
b1 = a11 * x1 + a12 * x2
b2 = a21 * x1 + a22 * x2

// Display the transformation
html`<div style="text-align: center; font-size: 1.3em; margin: 2em 0;">
  <p>
    ${tex`A = \begin{bmatrix} ${a11.toFixed(1)} & ${a12.toFixed(1)} \\ ${a21.toFixed(1)} & ${a22.toFixed(1)} \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} ${x1.toFixed(1)} \\ ${x2.toFixed(1)} \end{bmatrix}, \quad \mathbf{b} = A\mathbf{x} = \begin{bmatrix} ${b1.toFixed(1)} \\ ${b2.toFixed(1)} \end{bmatrix}`}
  </p>
</div>`

gridPoints = {
  const points = [];
  for (let i = -4; i <= 4; i += 0.5) {
    for (let j = -4; j <= 4; j += 0.5) {
      const DomainX = i;
      const DomainY = j;
      const transformedX = a11 * i + a12 * j;
      const transformedY = a21 * i + a22 * j;
      points.push({
        origX: DomainX,
        origY: DomainY,
        transX: transformedX,
        transY: transformedY,
        isGrid: true
      });
    }
  }
  return points;
}

// Create unit vectors and their transformations
unitVectors = [
  // Standard basis vectors
  {origX: 0, origY: 0, transX: 0, transY: 0, type: "origin"},
  {origX: 1, origY: 0, transX: a11, transY: a21, type: "e1"},
  {origX: 0, origY: 1, transX: a12, transY: a22, type: "e2"},
  // Input vector
  {origX: 0, origY: 0, transX: 0, transY: 0, type: "origin"},
  {origX: x1, origY: x2, transX: b1, transY: b2, type: "vector"}
]

// Create the plot
Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 800,
  height: 400,
  grid: true,
  facet: {data: [{side: "Domain"}, {side: "Codomain"}], x: "side"},
  fx: {label: null, domain: ["Domain", "Codomain"]},
  x: {label: "x_1", domain: [-8, 8]},
  y: {label: "x_2", domain: [-8, 8]},
  marks: [
    // Grid axes
    Plot.ruleY([0], {stroke: "#ddd"}),
    Plot.ruleX([0], {stroke: "#ddd"}),
    
    // Grid points (Domain)
    Plot.dot(gridPoints, {
      x: "origX", 
      y: "origY", 
      fx: () => "Domain",
      r: 1,
      fill: "#ccc",
      fillOpacity: 0.7
    }),
    
    // Grid points (Codomain)
    Plot.dot(gridPoints, {
      x: "transX", 
      y: "transY", 
      fx: () => "Codomain",
      r: 1,
      fill: "#ccc",
      fillOpacity: 0.7
    }),
    
    // Unit vector e1 (Domain)
    Plot.arrow([{x1: 0, y1: 0, x2: 1, y2: 0}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Domain",
      stroke: "orange",
      strokeWidth: 2,
      fill: "orange"
    }),
    
    // Unit vector e2 (Domain)
    Plot.arrow([{x1: 0, y1: 0, x2: 0, y2: 1}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Domain",
      stroke: "purple",
      strokeWidth: 2,
      fill: "purple"
    }),
    
    // Input vector x (Domain)
    Plot.arrow([{x1: 0, y1: 0, x2: x1, y2: x2}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Domain",
      stroke: "steelblue",
      strokeWidth: 3,
      fill: "steelblue"
    }),
    
    // Codomain unit vector e1
    Plot.arrow([{x1: 0, y1: 0, x2: a11, y2: a21}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Codomain",
      stroke: "orange",
      strokeWidth: 2,
      fill: "orange"
    }),
    
    // Codomain unit vector e2
    Plot.arrow([{x1: 0, y1: 0, x2: a12, y2: a22}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Codomain",
      stroke: "purple",
      strokeWidth: 2,
      fill: "purple"
    }),
    
    // Codomain vector b
    Plot.arrow([{x1: 0, y1: 0, x2: b1, y2: b2}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      fx: () => "Codomain",
      stroke: "crimson",
      strokeWidth: 3,
      fill: "crimson"
    }),
    
    // Labels
    Plot.text([
      {x: 1, y: 0.2, text: "e₁", fx: "Domain"},
      {x: 0.2, y: 1, text: "e₂", fx: "Domain"},
      {x: x1/2, y: x2/2 + 0.3, text: "x", fx: "Domain"},
      {x: a11, y: a21 + 0.2, text: "Ae₁", fx: "Codomain"},
      {x: a12 + 0.2, y: a22, text: "Ae₂", fx: "Codomain"},
      {x: b1/2, y: b2/2 + 0.3, text: "b=Ax", fx: "Codomain"}
    ], {
      x: "x", y: "y", 
      fx: "fx",
      text: "text",
      fontSize: 12,
      fontWeight: "bold"
    })
  ]
})