Matrices and Linear Maps

A matrix is a rectangular array of numbers that represents a linear map (also called a linear transformation). This section explores how matrices encode linear transformations, how basis vectors determine matrix columns, and how matrix operations correspond to geometric operations.

Matrices Represent Linear Maps

Every linear transformation in the plane can be represented by a 2×2 matrix. If $A$ is a linear transformation with matrix $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$, then for any vector $\mathbf{X} = \lbrack x, y \rbrack^T$:

\[A(\mathbf{X}) = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} px + qy \\ rx + sy \end{bmatrix}\]

This formula is fundamental: the matrix acts on a vector to produce a new vector.

The Identity Transformation

The identity transformation, denoted $I$, sends every vector into itself:

\[I(\mathbf{X}) = \mathbf{X}, \quad \text{for every vector } \mathbf{X}\]

Since $I$ sends $\mathbf{X} = \lbrack x, y \rbrack^T$ into itself, the system is:

\[\begin{aligned} x' &= x \\ y' &= y \end{aligned}\]

The matrix of the identity transformation is:

\[m(I) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]

The Zero Transformation

The zero transformation, denoted $0$, sends every vector into the zero vector:

\[0(\mathbf{X}) = \mathbf{0}, \quad \text{for all } \mathbf{X}\]

The matrix of the zero transformation is:

\[m(0) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\]

Origin is Always Fixed

Every linear transformation sends the origin to the origin:

\[A(\mathbf{0}) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \mathbf{0}\]

Worked Example: Stretching

Let $D$ be stretching by factor 2. Then:

\[\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = D\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x \\ 2y \end{bmatrix}\]

Worked Example: Projection

Let $P$ be projection onto the line along $\lbrack 1, 2 \rbrack^T$. Then:

\[\begin{bmatrix} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{4}{5} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = P\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{1}{5}x + \frac{2}{5}y \\ \frac{2}{5}x + \frac{4}{5}y \end{bmatrix}\]

Matrix as Image of Basis Vectors

A powerful insight is that the columns of a transformation matrix are the images of the standard basis vectors. Recall from Points and Coordinates that the standard basis vectors are $\mathbf{e}_1 = \lbrack 1, 0 \rbrack^T$ and $\mathbf{e}_2 = \lbrack 0, 1 \rbrack^T$.

If $A$ has matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then:

\[A(\mathbf{e}_1) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix}\]

\[A(\mathbf{e}_2) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} b \\ d \end{bmatrix}\]

Key insight: The first column of the matrix is the image of $\mathbf{e}_1$, and the second column is the image of $\mathbf{e}_2$.

Deriving the Projection Matrix

We can derive transformation matrices by computing what happens to the basis vectors. For orthogonal projection onto a line along vector $\lbrack u, v \rbrack$:

function projection_matrix_symbolic()
    @variables u v
    [orthproj([u, v], [1, 0])[1] orthproj([u, v], [0, 1])[1];
     orthproj([u, v], [1, 0])[2] orthproj([u, v], [0, 1])[2]]
end

julia> projection_matrix_symbolic()
2×2 Matrix{Num}:
 (u^2) / (u^2 + v^2)  (u*v) / (u^2 + v^2)
 (u*v) / (u^2 + v^2)  (v^2) / (u^2 + v^2)

Substituting specific values:

function projection_matrix(x::Vector)
    @variables u v
    Symbolics.value.(substitute.(projection_matrix_symbolic(), 
                                  (Dict(u => x[1], v => x[2]),)))
end

julia> x = [2, 5]
julia> P = projection_matrix(x)
2×2 Matrix{Float64}:
 0.137931  0.344828
 0.344828  0.862069

# Projecting the line onto itself returns itself
julia> P * x
2-element Vector{Float64}:
 2.0
 5.0

# Projecting another vector
julia> P * [1, 2]
2-element Vector{Float64}:
 0.8275862068965518
 2.068965517241379

Projection Using Polar Coordinates

Using a unit vector $\lbrack \cos\theta, \sin\theta \rbrack$:

function projection_matrix_symbolic_polar()
    @variables u v θ
    simplify.(substitute.(projection_matrix_symbolic(),
                          (Dict(u => cos(θ), v => sin(θ)),)))
end

julia> projection_matrix_symbolic_polar()
2×2 Matrix{Num}:
       cos(θ)^2  (1//2)*sin(2θ)
 (1//2)*sin(2θ)        sin(θ)^2

function projection_matrix_polar(n::Number)
    @variables u v θ
    Symbolics.value.(substitute.(projection_matrix_symbolic_polar(), θ => deg2rad(n)))
end

julia> P = projection_matrix_polar(45)
2×2 Matrix{Float64}:
 0.5  0.5
 0.5  0.5

# Projecting onto the 45° line always gives a point on that line
julia> round.(P * [1, 2], digits = 2)
2-element Vector{Float64}:
 1.5
 1.5

julia> round.(P * [3, 7], digits = 2)
2-element Vector{Float64}:
 5.0
 5.0

Deriving the Rotation Matrix

For a rotation by angle $\theta$ (see Linear Transformations for the rotation formula):

function rotation_matrix_symbolic()
    @variables θ
    # Calculate for e₁ = [1, 0]
    x′ = (cos(θ) * 1) - (sin(θ) * 0)
    y′ = (sin(θ) * 1) + (cos(θ) * 0)
    # Calculate for e₂ = [0, 1]
    x′′ = (cos(θ) * 0) - (sin(θ) * 1)
    y′′ = (sin(θ) * 0) + (cos(θ) * 1)
    [x′ x′′; y′ y′′]
end

julia> rotation_matrix_symbolic()
2×2 Matrix{Num}:
 cos(θ)  -sin(θ)
 sin(θ)   cos(θ)

function rotation_matrix(d::Number)
    @variables θ
    Symbolics.value.(substitute.(rotation_matrix_symbolic(), θ => deg2rad(d)))
end

julia> R = rotation_matrix(45)
2×2 Matrix{Float64}:
 0.707107  -0.707107
 0.707107   0.707107

julia> R * [2, 5]
2-element Vector{Float64}:
 -2.121320343559642
  4.949747468305833

Change of Basis

A change of basis transforms coordinates from one coordinate system to another. The standard basis $\lbrack \mathbf{e}_1, \mathbf{e}_2 \rbrack$ defines the unit square, where any vector $\mathbf{v}$ is expressed as:

\[\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2\]

Linear Maps as Change of Basis

A linear map transforms the standard basis to a new basis $\lbrack \mathbf{a}_1, \mathbf{a}_2 \rbrack$. The image of vector $\mathbf{v}$ becomes:

\[\mathbf{v}' = v_1\mathbf{a}_1 + v_2\mathbf{a}_2\]

This is called a linear combination. The transformation duplicates the geometry of the $\lbrack \mathbf{e}_1, \mathbf{e}_2 \rbrack$-system in the $\lbrack \mathbf{a}_1, \mathbf{a}_2 \rbrack$-system.

Important observation: The components of $\mathbf{v}'$ with respect to the standard basis are different from its components with respect to the new basis. In the $\lbrack \mathbf{a}_1, \mathbf{a}_2 \rbrack$-system, the vector $\mathbf{v}'$ has the same components as $\mathbf{v}$ had in the standard system.

Worked Example: Change of Basis

Let the new basis be:

\[\mathbf{a}_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad \mathbf{a}_2 = \begin{bmatrix} -2 \\ 4 \end{bmatrix}\]

For vector $\mathbf{v} = \lbrack 1/2, 1 \rbrack^T$ in the standard system:

\[\mathbf{v}' = \frac{1}{2} \times \begin{bmatrix} 2 \\ 1 \end{bmatrix} + 1 \times \begin{bmatrix} -2 \\ 4 \end{bmatrix} = \begin{bmatrix} -1 \\ 9/2 \end{bmatrix}\]

julia> a₁ = [2, 1]
julia> a₂ = [-2, 4]
julia> v = [1/2, 1]

julia> v′ = (v[1] * a₁) + (v[2] * a₂)
2-element Vector{Float64}:
 -1.0
  4.5

The vector $\mathbf{v}' = \lbrack -1, 4.5 \rbrack^T$ has coordinates in the standard system. But with respect to the $\lbrack \mathbf{a}_1, \mathbf{a}_2 \rbrack$-system, it has coordinates $\lbrack 1/2, 1 \rbrack^T$ — the same as the original $\mathbf{v}$.

Transforming Basis Vectors

What happens when we transform the standard basis vectors themselves?

julia> v = [1, 0]  # This is e₁
julia> v′ = (v[1] * a₁) + (v[2] * a₂)
2-element Vector{Int64}:
 2
 1

julia> v = [0, 1]  # This is e₂
julia> v′ = (v[1] * a₁) + (v[2] * a₂)
2-element Vector{Int64}:
 -2
  4

This confirms that $\mathbf{e}_1 \mapsto \mathbf{a}_1$ and $\mathbf{e}_2 \mapsto \mathbf{a}_2$, illustrating why the matrix columns are the images of the basis vectors.

Determinants

The determinant measures how a linear map changes areas. A linear map takes the unit square spanned by $\lbrack \mathbf{e}_1, \mathbf{e}_2 \rbrack$ (area = 1) to a parallelogram spanned by $\lbrack \mathbf{a}_1, \mathbf{a}_2 \rbrack$.

Computing the Determinant

For a matrix $A$ with columns $\mathbf{a}_1 = \lbrack a_{1,1}, a_{2,1} \rbrack^T$ and $\mathbf{a}_2 = \lbrack a_{1,2}, a_{2,2} \rbrack^T$:

\[\lvert A \rvert = \begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{vmatrix} = a_{1,1}a_{2,2} - a_{1,2}a_{2,1}\]

Geometric Derivation

The area of a parallelogram spanned by two vectors can be found using the triangle formula. If $T$ is the area of the triangle formed by $\mathbf{a}_1$ and $\mathbf{a}_2$:

\[T = \frac{1}{2}a_{1,1}a_{2,2} - \frac{1}{2}a_{1,2}a_{2,1}\]

Since the parallelogram area $P = 2T$:

\[P = a_{1,1}a_{2,2} - a_{1,2}a_{2,1}\]

This is exactly the determinant formula.

Geometric Interpretation of Determinant Values

The determinant characterizes how a linear map affects geometry:

• If $\lvert A \rvert = 1$: the map preserves areas
• If $0 \leq \lvert A \rvert < 1$: the map shrinks areas
• If $\lvert A \rvert = 0$: the matrix is rank deficient (collapses space to a line or point)
• If $\lvert A \rvert > 1$: the map expands areas
• If $\lvert A \rvert < 0$: the map reverses orientation (in addition to any area change)

Worked Example: Determinant Calculation

For the change of basis matrix $A = \begin{bmatrix} 2 & -2 \\ 1 & 4 \end{bmatrix}$:

\[\lvert A \rvert = (2)(4) - (-2)(1) = 8 + 2 = 10\]

This means the linear map expands areas by a factor of 10.

Composing Linear Maps

Composing two linear transformations corresponds to matrix multiplication. If we first apply transformation $A$ to get $\mathbf{v}'$, then apply transformation $B$ to get $\mathbf{v}''$:

\[\mathbf{v}' = A\mathbf{v}, \quad \mathbf{v}'' = B\mathbf{v}'\]

Deriving the Product Matrix

Starting with:

\[\mathbf{v}' = \begin{bmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} a_{1,1}v_1 + a_{1,2}v_2 \\ a_{2,1}v_1 + a_{2,2}v_2 \end{bmatrix}\]

Then applying $B$:

\[\mathbf{v}'' = \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \end{bmatrix} \begin{bmatrix} a_{1,1}v_1 + a_{1,2}v_2 \\ a_{2,1}v_1 + a_{2,2}v_2 \end{bmatrix}\]

Collecting terms in $v_1$ and $v_2$:

\[\mathbf{v}'' = \begin{bmatrix} b_{1,1}a_{1,1} + b_{1,2}a_{2,1} & b_{1,1}a_{1,2} + b_{1,2}a_{2,2} \\ b_{2,1}a_{1,1} + b_{2,2}a_{2,1} & b_{2,1}a_{1,2} + b_{2,2}a_{2,2} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\]

The product matrix $C = BA$ captures both transformations:

\[\mathbf{v}'' = B\mathbf{v}' = B(A\mathbf{v}) = BA\mathbf{v} = C\mathbf{v}\]

Matrix Multiplication Rule

The element $c_{i,j}$ is computed as the dot product of $B$'s $i$-th row and $A$'s $j$-th column.

Worked Example: Composing Transformations

julia> A = [-1 2; 0 3]
2×2 Matrix{Int64}:
 -1  2
  0  3

julia> B = [0 -2; -3 1]
2×2 Matrix{Int64}:
  0  -2
 -3   1

julia> v = [2, -1]

# Apply A, then B
julia> v′ = A * v
2-element Vector{Int64}:
 -4
 -3

julia> v′′ = B * v′
2-element Vector{Int64}:
 6
 9

# Same result using the product matrix
julia> C = B * A
2×2 Matrix{Int64}:
 0  -6
 3  -3

julia> C * v
2-element Vector{Int64}:
 6
 9

Matrix Multiplication is Not Commutative

Unlike multiplication of real numbers, matrix multiplication is not commutative:

\[AB \neq BA \quad \text{(in general)}\]

This reflects that the order of applying transformations matters geometrically.

Worked Example: Order Dependence

Consider a rotation by $-120°$ and a reflection about the $\mathbf{e}_1$-axis:

julia> A = [-0.5 0.866; -0.866 -0.5]  # Rotation by -120°
julia> B = [1 0; 0 -1]                 # Reflection about x-axis

julia> A * B  # Rotate, then reflect
2×2 Matrix{Float64}:
 -0.5    -0.866
 -0.866   0.5

julia> B * A  # Reflect, then rotate
2×2 Matrix{Float64}:
 -0.5    0.866
  0.866  0.5

The results are different matrices, confirming that linear map composition is order dependent.

Summary

See Also

• Points and Coordinates — Standard basis vectors
• Linear Transformations — Types of transformations (rotation, projection, reflection, scaling)