Linear Transformations

A linear transformation is a rule that assigns to each vector in the plane another vector in the plane, preserving the operations of vector addition and scalar multiplication. We write $T(\mathbf{X})$ for the vector that $T$ assigns to $\mathbf{X}$, called the image of $\mathbf{X}$.

What Are Linear Transformations?

A transformation of the plane maps each vector $\mathbf{X} = \lbrack x, y \rbrack^T$ to a new vector $\mathbf{X}' = \lbrack x', y' \rbrack^T$. A transformation $A$ is linear if it can be expressed as:

\[\begin{cases} x' = ax + by \\ y' = cx + dy \end{cases}\]

for some scalars $a, b, c, d$. The symbol

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

is called the matrix of $A$, denoted $m(A)$.

Key Examples of Linear Transformations

The following transformations are all linear:

• Projections onto lines through the origin: $m(P) = \begin{bmatrix} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{4}{5} \end{bmatrix}$
• Reflections through lines through the origin: $m(S) = \begin{bmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{bmatrix}$
• Scalings (stretching): $m(D_r) = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix}$
• Rotations: $m(R_\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

Properties of Linear Transformations

Linear transformations have several fundamental geometric properties:

Origin Fixed

The origin is always mapped to itself. If $\mathbf{X} = \lbrack 0, 0 \rbrack^T$, then:

\[\mathbf{X}' = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\]

Lines Map to Lines

Straight lines are preserved under linear transformations. A line through the origin remains a line through the origin; a line not through the origin maps to another line (which may or may not pass through the origin).

Parallelism Preserved

Parallel lines remain parallel after transformation. If two lines are parallel before the transformation, their images are also parallel.

Effect on Standard Basis

Understanding how a linear transformation acts on the standard basis vectors $\mathbf{e}_1 = \lbrack 1, 0 \rbrack^T$ and $\mathbf{e}_2 = \lbrack 0, 1 \rbrack^T$ completely determines the transformation. The columns of the transformation matrix are precisely the images of these basis vectors:

\[\mathbf{a}_1 = A\mathbf{e}_1 = \begin{bmatrix} a \\ c \end{bmatrix}, \quad \mathbf{a}_2 = A\mathbf{e}_2 = \begin{bmatrix} b \\ d \end{bmatrix}\]

This means $A = \lbrack \mathbf{a}_1 \quad \mathbf{a}_2 \rbrack$.

Scaling

Scaling (also called stretching) multiplies each coordinate by a factor. Let $D_r$ be the transformation that sends each vector into $r$ times itself:

\[D_r(\mathbf{X}) = r\mathbf{X}\]

Uniform Scaling Matrix

For uniform scaling by factor $r$:

\[\mathbf{v}' = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix} \mathbf{v} = \begin{bmatrix} rv_1 \\ rv_2 \end{bmatrix}\]

Geometric Effect

• If $\lvert r \rvert > 1$: vectors are enlarged
• If $\lvert r \rvert < 1$: vectors are shortened
• If $r < 0$: additionally, direction is reversed

Non-Uniform Scaling

Non-uniform scaling allows different scaling factors along each axis:

\[\mathbf{v}' = \begin{bmatrix} a_{1,1} & 0 \\ 0 & a_{2,2} \end{bmatrix} \mathbf{v}\]

For example, with $a_{1,1} = \frac{1}{2}$ and $a_{2,2} = 2$, objects are compressed horizontally and stretched vertically.

Effect on Area

Scaling affects area by a factor of $a_{1,1} \cdot a_{2,2}$. The action ellipse (the image of the unit circle under the transformation) reveals this distortion—for uniform scaling it remains a circle; for non-uniform scaling it becomes an ellipse.

Scaling matrices are diagonal matrices, where all off-diagonal elements are zero.

Reflection

Reflection maps each vector to its mirror image about a line through the origin.

Reflection About Coordinate Axes

Reflection about the $\mathbf{e}_1$-axis (x-axis):

\[\mathbf{v}' = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \mathbf{v} = \begin{bmatrix} v_1 \\ -v_2 \end{bmatrix}\]

This changes the sign of the second component, flipping the vector vertically.

Reflection About the Line $x_2 = x_1$

\[\mathbf{v}' = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \mathbf{v} = \begin{bmatrix} v_2 \\ v_1 \end{bmatrix}\]

This interchanges the components of the input vector.

General Reflection Formula

Given a vector $\mathbf{X}$ and a direction vector $\mathbf{U}$, the reflection $S(\mathbf{X})$ about the line along $\mathbf{U}$ is computed using the orthogonal projection:

\[S(\mathbf{X}) = 2P_{\mathbf{U}}(\mathbf{X}) - \mathbf{X}\]

where $P_{\mathbf{U}}(\mathbf{X})$ is the projection of $\mathbf{X}$ onto $\mathbf{U}$.

function reflection(X::Vector, U::Vector)
    P = orthproj(U, X)  # Project X onto U
    2 * P - X
end

julia> X = [0, 10]
2-element Vector{Int64}:
  0
 10

julia> U = [1, 2]
2-element Vector{Int64}:
 1
 2

julia> reflection(X, U)
2-element Vector{Float64}:
 8.0
 6.0

Geometric Properties

• Reflections do not change the magnitude of area, but they change the sign of area (orientation is reversed)
• Rotating $\mathbf{e}_1$ into $\mathbf{e}_2$ is counterclockwise; after reflection, the corresponding rotation is clockwise
• The action ellipse of any reflection is a circle

Rotation

Rotation turns vectors around the origin by a fixed angle while preserving their length.

Rotation by 180°

The matrix

\[\mathbf{v}' = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \mathbf{v}\]

rotates vectors by $180°$. Although it appears similar to a reflection, rotating $\mathbf{a}_1$ into $\mathbf{a}_2$ moves counterclockwise, confirming it is a rotation.

Deriving the Rotation Matrix

To rotate the standard basis vector $\mathbf{e}_1 = \lbrack 1, 0 \rbrack^T$ by angle $\alpha$ counterclockwise:

\[\mathbf{e}_1' = \begin{bmatrix} \cos\alpha \\ \sin\alpha \end{bmatrix}\]

Similarly, $\mathbf{e}_2$ rotates to:

\[\mathbf{e}_2' = \begin{bmatrix} -\sin\alpha \\ \cos\alpha \end{bmatrix}\]

This gives the rotation matrix:

\[R_\alpha = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\]

Rotation of Arbitrary Vectors

For any vector $\mathbf{X} = \lbrack x, y \rbrack^T$ with polar angle $\phi$:

\[\mathbf{X} = \lvert\mathbf{X}\rvert \begin{bmatrix} \cos\phi \\ \sin\phi \end{bmatrix}\]

After rotation by $\theta$, the new polar angle is $\phi + \theta$ (see Polar Form for details), and:

\[R_\theta(\mathbf{X}) = \lvert\mathbf{X}\rvert \begin{bmatrix} \cos(\phi + \theta) \\ \sin(\phi + \theta) \end{bmatrix} = \begin{bmatrix} x\cos\theta - y\sin\theta \\ x\sin\theta + y\cos\theta \end{bmatrix}\]

Verification via Dot Product

We can verify that $R$ rotates by angle $\alpha$ using the geometric formula for the dot product:

\[\mathbf{v} \cdot \mathbf{v}' = \lVert\mathbf{v}\rVert^2 \cos\alpha\]

For $\mathbf{v}' = R_\alpha \mathbf{v}$:

\[\begin{aligned} \mathbf{v} \cdot \mathbf{v}' &= v_1(v_1\cos\alpha - v_2\sin\alpha) + v_2(v_1\sin\alpha + v_2\cos\alpha) \\ &= v_1^2\cos\alpha - v_1 v_2\sin\alpha + v_1 v_2\sin\alpha + v_2^2\cos\alpha \\ &= (v_1^2 + v_2^2)\cos\alpha = \lVert\mathbf{v}\rVert^2 \cos\alpha \end{aligned}\]

Julia Implementation

function rotation(θ::Number, v::Vector)
    x′ = cos(deg2rad(θ)) * v[1] - sin(deg2rad(θ)) * v[2]
    y′ = sin(deg2rad(θ)) * v[1] + cos(deg2rad(θ)) * v[2]
    [round(x′), round(y′)]
end

julia> v = [5, 0]
2-element Vector{Int64}:
 5
 0

julia> rotation(90, v)
2-element Vector{Float64}:
 0.0
 5.0

julia> rotation(-90, v)
2-element Vector{Float64}:
  0.0
 -5.0

julia> w = rotation(45, v)
2-element Vector{Float64}:
 4.0
 4.0

Rotation Properties

Rotations are rigid body motions—they preserve both lengths and angles. A rotation matrix is an orthogonal matrix, meaning its column vectors are orthonormal (unit length and mutually perpendicular). The action ellipse of a rotation is always a circle, and rotations preserve area.

For a $45°$ rotation:

\[R_{45°} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\]

Shear

A shear transformation maps rectangles to parallelograms while preserving one coordinate axis.

Shear Along the $\mathbf{e}_1$-Axis

The shear matrix:

\[\mathbf{v}' = \begin{bmatrix} 1 & d_1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} v_1 + v_2 d_1 \\ v_2 \end{bmatrix}\]

This shifts points horizontally by an amount proportional to their vertical position. The $v_2$-coordinate remains unchanged.

Shear Along the $\mathbf{e}_2$-Axis

\[\mathbf{v}' = \begin{bmatrix} 1 & 0 \\ d_2 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_1 d_2 + v_2 \end{bmatrix}\]

This shifts points vertically by an amount proportional to their horizontal position.

Geometric Effect

The shear transformation converts a unit circle (the "Phoenix" shape) into an ellipse. For example, with $d_1 = 0.5$:

• The unit vector $\mathbf{e}_2 = \lbrack 0, 1 \rbrack^T$ maps to $\lbrack 0.5, 1 \rbrack^T$
• The shape tilts while maintaining the horizontal axis

Special Case: Projecting onto an Axis

To find a shear that maps $\mathbf{v} = \lbrack v_1, v_2 \rbrack^T$ to $\mathbf{v}' = \lbrack v_1, 0 \rbrack^T$:

\[\mathbf{v}' = \begin{bmatrix} 1 & 0 \\ -v_2/v_1 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\]

Properties

• Shears do not change areas—a rectangle and its image parallelogram have equal area (same base, same height)
• One practical application is in font design, where italic characters are created by shearing regular characters

Projection

A projection (specifically, a parallel projection) reduces dimensionality by mapping vectors onto a subspace, like sunlight casting shadows.

Orthogonal vs. Oblique Projection

• Orthogonal projection: angle of incidence with the target line is 90°
• Oblique projection: any other angle

Orthogonal projections are fundamental in linear algebra—they provide the best approximation in a subspace (see Projections and Distance for details).

Projection onto the $\mathbf{e}_1$-Axis

\[\mathbf{v}' = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \mathbf{v} = \begin{bmatrix} v_1 \\ 0 \end{bmatrix}\]

This "flattens" any vector onto the x-axis by setting its $v_2$-coordinate to zero.

General Projection Matrix

To project onto a line defined by unit vector $\mathbf{u}$, the projection matrix is:

\[A = \frac{\mathbf{u}\mathbf{u}^T}{\lVert\mathbf{u}\rVert^2}\]

This is a dyadic matrix formed from the outer product of $\mathbf{u}$ with itself. The projection of $\mathbf{x}$ onto $\mathbf{u}$ is:

\[A\mathbf{x} = \frac{\mathbf{u}\mathbf{u}^T}{\lVert\mathbf{u}\rVert^2} \mathbf{x} = \frac{(\mathbf{u} \cdot \mathbf{x})}{\lVert\mathbf{u}\rVert^2} \mathbf{u}\]

This matches the projection formula from Projections and Distance.

Julia Implementation

function projection_matrix(u::Vector)
    u * transpose(u) ./ (transpose(u) * u)
end

julia> u = [1, 2]
2-element Vector{Int64}:
 1
 2

julia> A = projection_matrix(u)
2×2 Matrix{Float64}:
 0.2  0.4
 0.4  0.8

julia> v = [1, 2]
2-element Vector{Int64}:
 1
 2

julia> v′ = A * v
2-element Vector{Float64}:
 1.0
 2.0

Idempotence

Projection matrices are idempotent: $A = AA$. Geometrically, once a vector is projected onto a line, applying the same projection again leaves it unchanged:

julia> A = projection_matrix([1, 2])
2×2 Matrix{Float64}:
 0.2  0.4
 0.4  0.8

julia> v = [1, 6]
2-element Vector{Int64}:
 1
 6

julia> v′ = A * v
2-element Vector{Float64}:
 2.6
 5.2

julia> A * v′
2-element Vector{Float64}:
 2.6
 5.2

Properties

• Projections reduce dimensionality: 2D vectors map to a 1D subspace
• The action ellipse of a projection is a line segment (traversed twice)
• Projections reduce area to zero—whatever the area was before, it becomes zero after projection

Summary of Transformations

Classification by Properties

Rigid Body Motions (preserve length and angle): Rotations, Reflections

Area-Preserving: Rotations, Reflections, Shears

Orthogonal Matrices (orthonormal columns): Rotations, Reflections

Diagonal Matrices: Uniform and non-uniform scaling

Idempotent Matrices ($A^2 = A$): Projections