Transformation Matrix: Types, Formulas & Examples

What a Transformation Matrix Is

A transformation matrix is a square matrix that maps the coordinates of a vector to new coordinates by matrix multiplication. When a transformation matrix ( T ) of order ( n \times n ) multiplies a column vector ( \mathbf{v} ) with ( n ) components, it produces a new vector ( \mathbf{v}' ) in the same space:

[ \mathbf{v}' = T \mathbf{v} ]

For a flat 2D plane, ( T ) is a ( 2 \times 2 ) matrix. For 3D space, it is ( 3 \times 3 ). The transformation acts on the whole coordinate system at once. A rotation matrix is a common example that turns without distorting.

The columns of ( T ) indicate where the basis vectors go.

How Do You Apply a Transformation Matrix to a Vector?

To apply a transformation matrix, place the vector's coordinates in a column and multiply on the left:

[ T \mathbf{v} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} ax + by \ cx + dy \end{bmatrix} ]

The order matters: the matrix sits on the left, the column vector on the right. For the full mechanics of multiplying, see multiplication of matrices.

The Types of Transformation Matrix

There are five transformations you will frequently encounter:

• Scaling: Enlarges or shrinks distances along the axes. Scale by ( k_x ) along x and ( k_y ) along y: [ \begin{bmatrix} k_x & 0 \ 0 & k_y \end{bmatrix} ]

• Rotation: Turns the figure about the origin by angle ( \theta ):
  [ \begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix} ]

• Shear: Slides one coordinate in proportion to the other:

  • A shear along x: [ \begin{bmatrix} 1 & k \ 0 & 1 \end{bmatrix} ]
  • A shear along y: [ \begin{bmatrix} 1 & 0 \ k & 1 \end{bmatrix} ]

• Reflection: Mirrors the figure across an axis:

  • In the x-axis: [ \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} ]
  • In the y-axis: [ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} ]
  • Across the line ( y = x ): [ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} ]

• Translation: Shifts a figure without rotating or resizing. It cannot be represented as a ( 2 \times 2 ) matrix and requires homogeneous coordinates: [ \begin{bmatrix} 1 & 0 & t_x \ 0 & 1 & t_y \ 0 & 0 & 1 \end{bmatrix} ]

What Are the Properties of a Transformation Matrix?

Properties:

• The determinant measures area change. For a ( 2 \times 2 ) matrix, ( \text{det} \ T ) indicates how area scales.
• A zero determinant collapses the figure. If ( \text{det} T = 0 ), the transformation cannot be undone.
• Invertible transformations have an inverse matrix. When ( \text{det} T \neq 0 ), the transformation can be reversed.
• Composition is matrix multiplication. Applying transformations is non-commutative, so the order matters.
• The identity matrix leaves everything fixed: [ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]

Examples of Transformation Matrix

Example 1

Apply the scaling matrix [ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} ] to the point ( (1,1) ).
[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \ 3 \end{bmatrix} ] Final answer: (2,3).

Example 2

Reflect ( (4,1) ) across the line ( y = x ).
Using the reflection matrix [ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} ]:
[ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 4 \ 1 \end{bmatrix} = \begin{bmatrix} 1 \ 4 \end{bmatrix} ]
Final answer: (1,4).

Example 3

Apply the shear [ \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} ] to the point ( (3,2) ).
[ \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \ 2 \end{bmatrix} = \begin{bmatrix} 7 \ 2 \end{bmatrix} ]
Final answer: (7,2).

Example 4

Find the constant ( a ) in the shear [ \begin{bmatrix} 1 & a \ 0 & 1 \end{bmatrix} ] that maps ( (3,2) ) to ( (7,2) ). From the equation ( 3 + 2a = 7 ):
[ 2a = 4 ightarrow a = 2 ]
Final answer: a=2.

Example 5

Transform the vector ( 5i + 4j ) using ( T = \begin{bmatrix} 2 & -3 \ 1 & 2 \end{bmatrix} ).
[ \begin{bmatrix} 2 & -3 \ 1 & 2 \end{bmatrix} \begin{bmatrix} 5 \ 4 \end{bmatrix} = \begin{bmatrix} -2 \ 13 \end{bmatrix} ]
Final answer: -2i + 13j.

Example 6

Compose a reflection in the x-axis with a scaling by 2, then apply to (1,3).
Combine matrices and apply: [ \begin{bmatrix} 2 & 0 \ 0 & -2 \end{bmatrix} \begin{bmatrix} 1 \ 3 \end{bmatrix} = \begin{bmatrix} 2 \ -6 \end{bmatrix} ]
Final answer: (2,-6).

Practice Questions on Transformation Matrix

1. Scale the point (2,5) by a factor of 3 along both axes.
2. Reflect the point (6,2) across the y-axis.
3. Apply the shear [ \begin{bmatrix} 1 & 3 \ 0 & 1 \end{bmatrix} ] to the point (2,4).
4. Compose a rotation by 90° with the shear [ \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} ] (shear first, then rotate), then apply it to (1,0).
5. Find the determinant of the reflection matrix [ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} ] and state what it tells you.

The Short Version

• A transformation matrix maps every point of a figure to a new location through a single matrix multiplication.
• The five core types are scaling, rotation, shear, reflection, and translation — each with its own matrix shape.
• Translation needs homogeneous coordinates (3×3 in 2D).
• The most common mistake is reflecting across ( y = x ) by negating coordinates instead of swapping them.