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Matrix Transformation — Definition, Formula & Examples

A matrix transformation is a function that takes a vector as input, multiplies it by a specific matrix, and produces a new vector as output. It can rotate, reflect, scale, or shear points in a coordinate plane or higher-dimensional space.

A matrix transformation T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m is a mapping defined by T(x)=AxT(\mathbf{x}) = A\mathbf{x}, where AA is an m×nm \times n matrix and x\mathbf{x} is a column vector in Rn\mathbb{R}^n. Every matrix transformation is a linear transformation, meaning it satisfies T(u+v)=T(u)+T(v)T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) and T(cu)=cT(u)T(c\mathbf{u}) = cT(\mathbf{u}) for all vectors u,v\mathbf{u}, \mathbf{v} and scalars cc.

Key Formula

T(x)=Ax=[a11a12a21a22][x1x2]=[a11x1+a12x2a21x1+a22x2]T(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 \\ a_{21}x_1 + a_{22}x_2 \end{bmatrix}
Where:
  • AA = The transformation matrix (here shown as 2×2)
  • x\mathbf{x} = The input column vector
  • T(x)T(\mathbf{x}) = The output vector after transformation

How It Works

To apply a matrix transformation, you multiply the transformation matrix AA by the input vector x\mathbf{x}. Each row of AA produces one component of the output vector through a dot product with x\mathbf{x}. For example, a 2×22 \times 2 matrix acting on a 2D vector yields another 2D vector — this is how geometric operations like rotation by 90°90° or reflection across the xx-axis are performed algebraically. The columns of AA tell you where the standard basis vectors e1,e2,…\mathbf{e}_1, \mathbf{e}_2, \ldots land after the transformation, which completely determines the function.

Worked Example

Problem: Apply the matrix transformation defined by A = [[0, -1], [1, 0]] to the vector x = [3, 5]ᵀ. What does this transformation do geometrically?
Step 1: Set up the matrix-vector multiplication.
T(x)=[0−110][35]T(\mathbf{x}) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 3 \\ 5 \end{bmatrix}
Step 2: Compute the first component using the dot product of row 1 with x.
(0)(3)+(−1)(5)=−5(0)(3) + (-1)(5) = -5
Step 3: Compute the second component using the dot product of row 2 with x.
(1)(3)+(0)(5)=3(1)(3) + (0)(5) = 3
Step 4: Identify the geometry: the standard basis vector e₁ = [1, 0]ᵀ maps to [0, 1]ᵀ and e₂ = [0, 1]ᵀ maps to [-1, 0]ᵀ. This is a 90° counterclockwise rotation.
T(x)=[−53]T(\mathbf{x}) = \begin{bmatrix} -5 \\ 3 \end{bmatrix}
Answer: The transformed vector is [-5, 3]ᵀ. The matrix performs a 90° counterclockwise rotation.

Another Example

Problem: Use the scaling matrix A = [[2, 0], [0, 3]] to transform the vector x = [4, -1]ᵀ.
Step 1: Multiply the matrix by the vector.
[2003][4−1]=[(2)(4)+(0)(−1)(0)(4)+(3)(−1)]\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} (2)(4) + (0)(-1) \\ (0)(4) + (3)(-1) \end{bmatrix}
Step 2: Simplify each component.
=[8−3]= \begin{bmatrix} 8 \\ -3 \end{bmatrix}
Answer: The output is [8, -3]ᵀ. The x-component was scaled by 2 and the y-component was scaled by 3.

Visualization

Why It Matters

Matrix transformations are central to high school and college linear algebra courses, where they connect abstract algebra to concrete geometry. In computer graphics and game development, every rotation, scaling, and projection on screen is computed using matrix transformations applied to vertex coordinates. They also appear in data science, where principal component analysis uses matrix transformations to reduce the dimensionality of datasets.

Common Mistakes

Mistake: Multiplying the vector by the matrix in the wrong order (writing xA instead of Ax).
Correction: Matrix-vector multiplication requires the matrix on the left and the column vector on the right. For an m × n matrix, the vector must have n rows. The order matters because matrix multiplication is not commutative.
Mistake: Confusing which dimension is the input and which is the output.
Correction: For an m × n matrix A, the input vector must be in ℝⁿ (n entries) and the output vector is in ℝᵐ (m entries). The number of columns of A must match the number of rows in x.

Related Terms