Mathematical Transformations

A mathematical transformation is a function T that maps elements from an initial set (domain) to another set (codomain), modifying inputs into outputs according to a strict mathematical rule.

In geometry and linear algebra, it maps vectors or points from one vector space to another, written as:

T : V → W

Here V is the domain, W is the Codomain and T is the function that maps V to W.

2. Linear vs. Non-Linear Transformations

The fundamental division between transformations relies on two algebraic rules:

• Additivity
  T(u + v) = T(u) + T(v)
• Homogeneity
  T(cu) = cT(u)

Linear Transformations

Linear transformations satisfy both conditions.

• They always map the origin to itself.
• T(0) = 0
• They map straight lines to straight lines.
• They keep grid lines straight and parallel.
• They can always be expressed as a matrix multiplication:
  T(x) = Ax

Non-Linear Transformations

Non-linear transformations fail at least one condition.

• They curve the underlying space.
• They warp grid lines.
• They move the origin.
• Examples include squaring coordinates:
  T(x, y) = (x², y)
• Or adding constants:
  T(x) = x + b
  (which is an affine transformation, not strictly linear).

Property	Linear Transformation	Non-Linear Transformation
Additivity	✔ Satisfied	✘ Violated
Homogeneity	✔ Satisfied	✘ Violated
Origin	Preserved	May Move
Grid Lines	Remain Straight	May Warp
Matrix Representation	Always Possible	Generally Not Possible

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3. Standard 2D Linear Geometric Transformations

In a two-dimensional Cartesian plane, standard linear transformations alter the spatial properties of shapes and can be written using a 2 × 2 matrix:

[ x′ ]  =  [ a  b ] [ x ]
[ y′ ]      [ c  d ] [ y ]

Note

Use arrow keys to adjust value.

0:00 / 1:01

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Scaling

Stretches or shrinks coordinates along axes.

sx	0
0	sy

Rotation

Rotates points counterclockwise around the origin by an angle θ.

cos θ	− sin θ
sin θ	cos θ

Shearing

Slants one coordinate axis parallel to another based on a factor k.

Horizontal Shear

1	k
0	1

Vertical Shear

1	0
k	1

Reflection

Flips the space across a line passing through the origin.

Across the x-axis:

1	0
0	−1

Note

Translation—sliding an object—is geometric but requires a 3 × 3 matrix with homogeneous coordinates to be handled linearly because it moves the origin.

Figure 3.1 — Examples of scaling, rotation, shearing, and reflection in two-dimensional space.

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4. Non-Geometric Transformations

Non-geometric transformations modify the properties or data structures of elements rather than their spatial positions, shapes, or orientations.

Transformation Type	Description	Examples
Data Transformations	Converting variable distributions in statistics.	Logarithmic transformation (log(x)) or Z-score normalization.
Color Space Transformations	Converting digital imagery pixels from one color representation to another.	RGB → CMYK or YCbCr color spaces.
Domain Transformations	Changing the functional domain of an equation.	Fourier Transform or Laplace Transform for mapping time-domain signals to the frequency domain.
Cryptographic Transformations	Permuting and substituting data bytes to encrypt plain text into ciphertext.	Algorithms such as AES.

Important Note

Unlike geometric transformations, non-geometric transformations change the underlying data, representation, or mathematical domain without necessarily altering the physical position or shape of objects.