What is the rotation matrix in R²?

The matrix for counterclockwise rotation by angle θ about the origin is [[cos θ, −sin θ], [sin θ, cos θ]]. It is orthogonal with determinant 1, preserving lengths, angles, and orientation. Rotations compose by adding angles: R_α · R_β = R_(α+β).

What is a reflection matrix?

A reflection matrix mirrors vectors across a line (in R²) or plane (in R³). Reflection across a line at angle α has matrix [[cos 2α, sin 2α], [sin 2α, −cos 2α]]. All reflection matrices are orthogonal, have determinant −1, and satisfy H² = I — reflecting twice returns to the original.

What is a shear transformation?

A shear displaces each point proportionally to its distance from a fixed line or plane. A horizontal shear in R² has matrix [[1, k], [0, 1]], shifting x-coordinates by k times the y-coordinate. Shears have determinant 1, so they preserve area despite distorting angles.

How does the determinant classify geometric transformations?

The absolute value |det(A)| gives the area or volume scaling factor. Positive determinant means orientation is preserved, negative means reversed. Determinant ±1 means area/volume is preserved (rotations, reflections, shears). Determinant 0 means the transformation collapses at least one dimension.

How do you combine geometric transformations?

Composing transformations corresponds to multiplying their matrices, applied right-to-left. Order matters: rotating then shearing differs from shearing then rotating. The SVD reveals that every linear transformation decomposes into a rotation, a coordinate-axis scaling, and another rotation.

Geometric Transformations: Matrices in R² & R³

How to Read a Transformation Matrix

The geometric effect of a matrix

A

is revealed by two pieces of information: what it does to the standard basis vectors (the columns) and what its determinant is.

Column

1

is the image of

\mathbf{e}_1 = (1, 0)

in

\mathbb{R}^2

or

(1, 0, 0)

in

\mathbb{R}^3

. Column

2

is the image of

\mathbf{e}_2

. The matrix maps the standard grid to the parallelogram (or parallelepiped) spanned by these column vectors.

The absolute value

|\det(A)|

measures how the transformation scales areas (in

\mathbb{R}^2

) or volumes (in

\mathbb{R}^3

). The sign of

\det(A)

indicates orientation: positive means the transformation preserves handedness, negative means it reverses it. An orthogonal matrix (

\det = \pm 1

) preserves all lengths and angles — it is a rigid motion.

Scaling

Uniform scaling multiplies every coordinate by the same factor:

T(\mathbf{x}) = c\mathbf{x}

, with matrix

cI

. When

c > 1

the transformation enlarges, when

0 < c < 1

it shrinks, and when

c < 0

it reflects through the origin and scales.

Non-uniform scaling stretches each axis independently. In

\mathbb{R}^2

,

T(x, y) = (c_1 x, c_2 y)

has matrix

\text{diag}(c_1, c_2)

. The horizontal axis is scaled by

c_1

and the vertical by

c_2

. A unit square maps to a rectangle with side lengths

|c_1|

and

|c_2|

.

The determinant is

c^n

for uniform scaling and

c_1 c_2

(or

c_1 c_2 c_3

) for non-uniform. When any scaling factor is zero, the transformation collapses that axis entirely and the determinant is zero.

In

\mathbb{R}^3

,

\text{diag}(c_1, c_2, c_3)

scales each coordinate axis independently. The unit cube maps to a rectangular box with side lengths

|c_1|

,

|c_2|

,

|c_3|

and volume

|c_1 c_2 c_3|

.

Rotations in R²

Rotation by angle

\theta

counterclockwise about the origin has matrix

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

The first column

R_\theta \mathbf{e}_1 = (\cos\theta, \sin\theta)

is the image of

(1, 0)

— the point on the unit circle at angle

\theta

. The second column

R_\theta \mathbf{e}_2 = (-\sin\theta, \cos\theta)

is the image of

(0, 1)

— the point at angle

\theta + 90°

.

The determinant is

\cos^2\theta + \sin^2\theta = 1

for every

\theta

: rotations preserve areas and orientation. The matrix is orthogonal:

R_\theta^T R_\theta = I

, so lengths and angles are preserved. The inverse is

R_\theta^{-1} = R_{-\theta} = R_\theta^T

— rotating backward by the same angle.

Rotations compose by adding angles:

R_\alpha R_\beta = R_{\alpha + \beta}

. This follows from the trigonometric addition formulas and corresponds to the fact that rotating by

\beta

then by

\alpha

is the same as rotating by

\alpha + \beta

.

Common cases:

R_{90°} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

,

R_{180°} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

,

R_{45°} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}

.

Rotations in R³

In three dimensions, a rotation is specified by an axis and an angle. Rotation by

\theta

about the

z

-axis leaves the

z

-coordinate unchanged and rotates the

xy

-plane:

R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Rotations about the

x

-axis and

y

-axis have the same

2 \times 2

rotation block embedded in different positions:

R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}, \quad R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}

Every

3 \times 3

rotation matrix is orthogonal with determinant

+1

. The axis of rotation is the eigenvector with eigenvalue

1

— the direction that remains fixed. Any rotation in

\mathbb{R}^3

can be decomposed into rotations about the coordinate axes (Euler angles), though the decomposition is not unique.

Axis of rotation	Matrix	Fixed direction	Plane that rotates
x-axis: Rₓ(θ)	[1 0 0; 0 cosθ −sinθ; 0 sinθ cosθ]	x-coordinate unchanged	yz-plane
y-axis: R_y(θ)	[cosθ 0 sinθ; 0 1 0; −sinθ 0 cosθ]	y-coordinate unchanged	xz-plane
z-axis: R_z(θ)	[cosθ −sinθ 0; sinθ cosθ 0; 0 0 1]	z-coordinate unchanged	xy-plane

Reflections in R²

Reflection across the

x

-axis flips the

y

-coordinate:

T(x, y) = (x, -y)

, with matrix

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

.

Reflection across the

y

-axis flips the

x

-coordinate: matrix

\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

.

Reflection across the line

y = x

swaps coordinates: matrix

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

.

Reflection across an arbitrary line through the origin at angle

\alpha

has matrix

H_\alpha = \begin{pmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{pmatrix}

All reflection matrices share the same properties: the determinant is

-1

(orientation-reversing, area-preserving), the matrix is orthogonal (lengths and angles preserved), and the matrix is involutory (

H^2 = I

) — reflecting twice returns every vector to its starting point. The eigenvalues are

+1

(vectors on the mirror line) and

-1

(vectors perpendicular to it).

Mirror line	Matrix	Coordinate effect
x-axis	[1 0; 0 −1]	(x, y) ↦ (x, −y)
y-axis	[−1 0; 0 1]	(x, y) ↦ (−x, y)
line y = x	[0 1; 1 0]	(x, y) ↦ (y, x) — swaps coordinates
line through origin at angle α	[cos 2α sin 2α; sin 2α −cos 2α]	general formula; specializes to the rows above for α = 0°, 90°, 45°

Reflections in R³

Reflection across a coordinate plane negates the coordinate perpendicular to that plane. Reflection across the

xy

-plane:

\text{diag}(1, 1, -1)

. Across the

xz

-plane:

\text{diag}(1, -1, 1)

. Across the

yz

-plane:

\text{diag}(-1, 1, 1)

.

Reflection across an arbitrary plane through the origin with unit normal

\mathbf{n}

is given by the Householder matrix:

H = I - 2\mathbf{n}\mathbf{n}^T

This matrix subtracts twice the component of each vector in the direction of

\mathbf{n}

, effectively mirroring across the plane perpendicular to

\mathbf{n}

. Householder reflections are orthogonal, have determinant

-1

, and satisfy

H^2 = I

. They are the building blocks of the QR decomposition and are widely used in numerical algorithms.

Projections

Orthogonal projection onto a subspace collapses each vector onto its nearest point in the subspace, discarding the perpendicular component.

Projection onto the

x

-axis in

\mathbb{R}^2

:

T(x, y) = (x, 0)

, with matrix

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}

.

Projection onto a line through the origin in direction

\mathbf{u}

:

P = \frac{\mathbf{u}\mathbf{u}^T}{\mathbf{u}^T\mathbf{u}}

Projection onto a plane with unit normal

\mathbf{n}

in

\mathbb{R}^3

:

P = I - \frac{\mathbf{n}\mathbf{n}^T}{\mathbf{n}^T\mathbf{n}}

All orthogonal projection matrices share the same algebraic signature:

P^2 = P

(idempotent — projecting twice is the same as projecting once),

P^T = P

(symmetric), eigenvalues are

0

and

1

, and

\text{rank}(P) = \text{tr}(P)

. The determinant is

0

unless the projection is onto the full space — projections always collapse at least one dimension.

Shears

A shear displaces each point in proportion to its distance from a fixed line or plane. In

\mathbb{R}^2

, a horizontal shear shifts the

x

-coordinate by

k

times the

y

-coordinate:

T(x, y) = (x + ky, \; y), \quad \text{matrix } \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

A vertical shear shifts the

y

-coordinate by

k

times the

x

-coordinate:

T(x, y) = (x, \; kx + y), \quad \text{matrix } \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}

Both are triangular matrices with determinant

1

: shears are area-preserving and orientation-preserving. They are not orthogonal — angles are distorted. A square sheared horizontally becomes a parallelogram of the same area but with tilted sides.

In

\mathbb{R}^3

, there are six possible shear directions (each coordinate shifted by a multiple of each other coordinate). Each is represented by an identity matrix with one off-diagonal entry replaced by

k

. Shears are fundamental building blocks — any invertible matrix with determinant

1

can be written as a product of shears.

Combining Transformations

Composing transformations corresponds to multiplying their matrices. A rotation followed by a scaling is

SA

where

S

is the scaling matrix and

A

is the rotation matrix. The product is applied right-to-left:

A

acts first, then

S

.

Order matters. Rotating then shearing produces a different result from shearing then rotating:

\text{Shear} \cdot \text{Rotation} \neq \text{Rotation} \cdot \text{Shear}

in general.

The singular value decomposition reveals the hidden geometric structure of any matrix:

A = U\Sigma V^T

, where

V^T

is a rotation (or rotation-reflection),

\Sigma

is a coordinate-axis scaling, and

U

is another rotation (or rotation-reflection). Every linear transformation is a rotation, followed by a scaling along the coordinate axes, followed by another rotation. The singular values in

\Sigma

measure the maximum stretching in each orthogonal direction.

This decomposition means no linear transformation is truly exotic — even the most complex-looking matrix is just three simple geometric operations composed together.

Determinant as Geometric Signature

The determinant classifies every linear transformation by its effect on size and orientation.

|\det(A)|

is the factor by which the transformation scales areas (in

\mathbb{R}^2

) or volumes (in

\mathbb{R}^3

). A unit square maps to a parallelogram of area

|\det(A)|

. A unit cube maps to a parallelepiped of volume

|\det(A)|

.

\det(A) > 0

: the transformation preserves orientation. Counterclockwise stays counterclockwise in

\mathbb{R}^2

; right-handed stays right-handed in

\mathbb{R}^3

. Rotations and shears fall in this category.

\det(A) < 0

: the transformation reverses orientation. Counterclockwise becomes clockwise; right-handed becomes left-handed. Reflections are the canonical example.

\det(A) = 0

: the transformation collapses at least one dimension. The image is a proper subspace — a line or point in

\mathbb{R}^2

, a plane, line, or point in

\mathbb{R}^3

. Projections onto proper subspaces and singular matrices fall here.

|\det(A)| = 1

: the transformation preserves area or volume. Rotations (

\det = +1

) and reflections (

\det = -1

) are the area-preserving and volume-preserving transformations. Shears also have

\det = 1

, preserving area despite distorting angles.

det(A)	Effect on area / volume	Effect on orientation	Typical transformations
\|det(A)\| > 1	expands by factor \|det(A)\|	preserves if det > 0, reverses if det < 0	uniform scaling with \|c\| > 1
0 < \|det(A)\| < 1	shrinks by factor \|det(A)\|	preserves if det > 0, reverses if det < 0	uniform scaling with 0 < \|c\| < 1
det(A) = +1	preserves area / volume	preserves orientation	rotation, shear
det(A) = −1	preserves area / volume	reverses orientation (handedness flip)	reflection, improper rotation
det(A) = 0	collapses at least one dimension (image is a proper subspace)	undefined / degenerate	projection onto a proper subspace, any singular matrix

Summary: Algebraic Signature of Each Family

Each family of geometric transformations has been treated in its own section, with its own matrix templates and its own algebraic identities. The table below sets all five side by side across the diagnostic signatures that distinguish them — orthogonality, determinant, eigenvalues, the characteristic algebraic identity, and what each family does to length and to area. It is the recognition card to keep nearby when a matrix is in hand and the question is which geometric family it belongs to.

Family	Orthogonal?	det(A)	Eigenvalues	Algebraic identity	Preserves length?	Preserves area/vol?
Scaling (uniform by c)	only when c = ±1	cⁿ	c (n-fold)	A = cI (diagonal)	only c = ±1	only c = ±1
Rotation R_θ	✓	+1	e^±iθ (complex pair; ℝ: 1 on rotation axis in ℝ³)	RᵀR = I; R⁻¹ = Rᵀ	✓	✓
Reflection H	✓	−1	+1 (on mirror) and −1 (perpendicular to mirror)	H² = I (involutory); Hᵀ = H	✓	✓
Projection P (orthogonal)	only when P = I	0 (proper); 1 (= I)	0 (perpendicular component) and 1 (in-subspace component)	P² = P (idempotent); Pᵀ = P	✗	✗
Shear	✗	+1	1 (repeated; defective — not diagonalizable)	triangular with 1's on the diagonal	✗	✓

Geometric Transformations

Rotations, Reflections, Projections, and More

How to Read a Transformation Matrix

Scaling

Rotations in R²

Rotations in R³

Reflections in R²

Reflections in R³

Projections

Shears

Combining Transformations

Determinant as Geometric Signature

Summary: Algebraic Signature of Each Family