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Linear Transformation — Definition, Formula & Examples

A linear transformation is a function between two vector spaces that preserves addition and scalar multiplication. In practice, every linear transformation from Rn\mathbb{R}^n to Rm\mathbb{R}^m can be represented by multiplying by an m×nm \times n matrix.

A function T:V→WT: V \to W between vector spaces VV and WW over a field FF is a linear transformation if for all vectors u,v∈V\mathbf{u}, \mathbf{v} \in V and all scalars c∈Fc \in F: (1) T(u+v)=T(u)+T(v)T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) and (2) T(cu)=cT(u)T(c\mathbf{u}) = cT(\mathbf{u}).

Key Formula

T(x)=AxT(\mathbf{x}) = A\mathbf{x}
Where:
  • TT = The linear transformation
  • AA = The m × n matrix representing the transformation
  • x\mathbf{x} = An input vector in \mathbb{R}^n

How It Works

To apply a linear transformation in Rn\mathbb{R}^n, you multiply a matrix AA by an input vector x\mathbf{x} to get the output T(x)=AxT(\mathbf{x}) = A\mathbf{x}. Each column of AA tells you where the corresponding standard basis vector lands after the transformation. For example, rotations, reflections, projections, and scaling are all linear transformations. To verify that a given function is linear, check that it satisfies both the addition and scalar multiplication properties — or equivalently, that T(cu+dv)=cT(u)+dT(v)T(c\mathbf{u} + d\mathbf{v}) = cT(\mathbf{u}) + dT(\mathbf{v}) for all vectors and scalars.

Worked Example

Problem: Let T be the linear transformation defined by the matrix A = [[2, 1], [0, 3]]. Find T(x) when x = [4, -1].
Step 1: Set up the matrix-vector multiplication.
T(x)=[2103][4−1]T(\mathbf{x}) = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -1 \end{bmatrix}
Step 2: Compute each entry of the result. The first entry is (2)(4) + (1)(-1) = 7. The second entry is (0)(4) + (3)(-1) = -3.
T(x)=[2(4)+1(−1)0(4)+3(−1)]=[7−3]T(\mathbf{x}) = \begin{bmatrix} 2(4) + 1(-1) \\ 0(4) + 3(-1) \end{bmatrix} = \begin{bmatrix} 7 \\ -3 \end{bmatrix}
Answer: T(x)=[7−3]T(\mathbf{x}) = \begin{bmatrix} 7 \\ -3 \end{bmatrix}

Why It Matters

Linear transformations are the central objects of study in linear algebra courses and appear throughout engineering, physics, and computer science. Computer graphics engines use them to rotate, scale, and project 3D models onto a 2D screen. In data science, techniques like principal component analysis (PCA) rely on linear transformations to reduce high-dimensional data.

Common Mistakes

Mistake: Assuming any function defined by a formula is linear — for instance, treating T(x) = x² or T(x) = x + 3 as linear transformations.
Correction: A linear transformation must satisfy T(0) = 0. Functions with constant terms (like x + 3) or nonlinear operations (like squaring) fail the linearity properties. Always verify both additivity and scalar multiplication.