Linear Transformation

57. Linear Transformation

A function $𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ between vector spaces is a linear transformation if it preserves vector addition and scalar multiplication.

57.1. The two axioms

For any vectors $\vec{𝑎}, \vec{𝑏} \in ℝ^{𝑛}$ and scalar $𝑐 \in ℝ$ :

1. Additivity

𝑇 (\vec{𝑎} + \vec{𝑏}) = 𝑇 (\vec{𝑎}) + 𝑇 (\vec{𝑏})

2. Homogeneity (scalar multiplication)

𝑇 (𝑐 \vec{𝑎}) = 𝑐 𝑇 (\vec{𝑎})

The two combine into a single condition: $𝑇 (𝑐 \vec{𝑎} + 𝑑 \vec{𝑏}) = 𝑐 𝑇 (\vec{𝑎}) + 𝑑 𝑇 (\vec{𝑏})$ for all scalars and vectors.

Example

$𝑇 : ℝ^{2} \to ℝ^{2}$ defined by $𝑇 ([\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}]) = [\begin{matrix} 2 𝑥 \\ 3 𝑦 \end{matrix}]$ is linear.

Check additivity: $\vec{𝑎} = [\begin{matrix} 1 \\ 2 \end{matrix}]$ , $\vec{𝑏} = [\begin{matrix} 3 \\ 1 \end{matrix}]$ :

\begin{matrix} 𝑇 (\vec{𝑎}) + 𝑇 (\vec{𝑏}) = [\begin{matrix} 2 \\ 6 \end{matrix}] + [\begin{matrix} 6 \\ 3 \end{matrix}] = [\begin{matrix} 8 \\ 9 \end{matrix}] \\ 𝑇 (\vec{𝑎} + \vec{𝑏}) = 𝑇 ([\begin{matrix} 4 \\ 3 \end{matrix}]) = [\begin{matrix} 8 \\ 9 \end{matrix}] ✓ \end{matrix}

Check homogeneity: $𝑐 = 3$ , $\vec{𝑎} = [\begin{matrix} 1 \\ 2 \end{matrix}]$ :

\begin{matrix} 𝑐 𝑇 (\vec{𝑎}) = 3 [\begin{matrix} 2 \\ 6 \end{matrix}] = [\begin{matrix} 6 \\ 18 \end{matrix}] \\ 𝑇 (𝑐 \vec{𝑎}) = 𝑇 ([\begin{matrix} 3 \\ 6 \end{matrix}]) = [\begin{matrix} 6 \\ 18 \end{matrix}] ✓ \end{matrix}

57.2. Consequences

Linearity forces $𝑇$ to be very rigid. From the two axioms:

$𝑇 (𝟎) = 𝟎$ (apply homogeneity with $𝑐 = 0$ )
$𝑇 (- \vec{𝑎}) = - 𝑇 (\vec{𝑎})$
$𝑇$ maps lines through the origin to lines through the origin (or to the origin itself)
$𝑇$ maps the origin to the origin
$𝑇$ maps parallelograms to parallelograms (or degenerate shapes)

57.3. Equivalent definition: matrix–vector product

Every linear transformation can be written as $𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}$ for some matrix $𝐴$ — see Matrix Representation. This means linear transformation $ℝ^{𝑛} \to ℝ^{𝑚}$ ⇔ $𝑚 \times 𝑛$ matrix.

57.4. Non-examples

$𝑇 (𝑥) = 𝑥 + 1$ : not linear (translates the origin off itself)
$𝑇 ([\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}]) = [\begin{matrix} 𝑥^{2} \\ 𝑦 \end{matrix}]$ : not linear (squaring isn’t homogeneous)
$𝑇 ([\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}]) = [\begin{matrix} 𝑥 𝑦 \\ 0 \end{matrix}]$ : not linear (product of components)

57.5. See also

Matrix–Vector Product — every matrix is a linear transformation
Matrix Representation — every linear transformation is a matrix
Image / Kernel — output range / null space
Composition — composing linear transformations
Rotation, Reflection, Scaling, Shear