1.3 Matrix Operations

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 \bold{x} \in \mathbb{R}^n a vector T(\bold{x}) \in \mathbb{R}^m.

Matrices serve as linear transformations

T(\bold{x}) = A\bold{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\bold{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} = \bold{b}

A\bold{x} = \bold{b}

To perform the matrix multiplication A\bold{x}, the number of columns in A must match the number of entries in the vector \bold{x}.
That is, if A is an m \times n matrix and \bold{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 \bold{x} = \begin{bmatrix} ${x1.toFixed(1)} \\ ${x2.toFixed(1)} \end{bmatrix}, \quad \bold{b} = A\bold{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"
    })
  ]
})

Addition and Subtraction

Multiplication by a Scalar

Multiplication by a Vector

Matrix-Matrix Multiplication