Linear Transformations and Their Properties

What is a linear transformation?

A function $T : V \to W$ between vector spaces is linear if for all $\mathbf{u}, \mathbf{v} \in V$ and scalars $c$:

1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$.
2. Homogeneity: $T(c \mathbf{v}) = c \, T(\mathbf{v})$.

Equivalently: $T(a \mathbf{u} + b \mathbf{v}) = a \, T(\mathbf{u}) + b \, T(\mathbf{v})$ (preserves linear combinations).

A consequence of these axioms: $T(\mathbf{0}) = \mathbf{0}$. Any map that doesn't send zero to zero is not linear (quick disqualifier).

Step-by-step — is $T(x, y) = (2x + y, x - y)$ linear?

Let $\mathbf{u} = (x_1, y_1)$ and $\mathbf{v} = (x_2, y_2)$, $c \in \mathbb{R}$.

Additivity check: $T(\mathbf{u} + \mathbf{v}) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2) + (y_1 + y_2), (x_1 + x_2) - (y_1 + y_2))$ $= (2 x_1 + y_1, x_1 - y_1) + (2 x_2 + y_2, x_2 - y_2) = T(\mathbf{u}) + T(\mathbf{v}) ✓$

Homogeneity check: $T(c \mathbf{u}) = T(c x_1, c y_1) = (2 c x_1 + c y_1, c x_1 - c y_1) = c (2 x_1 + y_1, x_1 - y_1) = c \, T(\mathbf{u}) ✓$

Both axioms hold: $T$ is linear.

Matrix representation

Any linear $T : \mathbb{R}^n \to \mathbb{R}^m$ is given by matrix multiplication: $T(\mathbf{x}) = A \mathbf{x}$. The matrix is built column by column: $A = [T(\mathbf{e}_1), T(\mathbf{e}_2), \ldots, T(\mathbf{e}_n)]$

where $\mathbf{e}_j$ is the $j$-th standard basis vector.

For our $T$:

• $T(\mathbf{e}_1) = T(1, 0) = (2 \cdot 1 + 0, 1 - 0) = (2, 1)$.
• $T(\mathbf{e}_2) = T(0, 1) = (0 + 1, 0 - 1) = (1, -1)$.

$A = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix}$

Check: $T(x, y) = A (x, y)^T = (2x + y, x - y)$ ✓.

What the matrix does geometrically

$A$ takes:

• $(1, 0)$ to $(2, 1)$ — the x-axis tilts up-right.
• $(0, 1)$ to $(1, -1)$ — the y-axis tilts down-right.
• $(1, 1)$ to $(3, 0)$ — the square's diagonal corner stretches horizontally.

The unit square with corners $\mathbf{0}, (1, 0), (1, 1), (0, 1)$ maps to a parallelogram with corners $\mathbf{0}, (2, 1), (3, 0), (1, -1)$.

Area scaling: $|\det A| = |2(-1) - 1 \cdot 1| = 3$. So the parallelogram has 3 times the area of the unit square.

Orientation: $\det A = -3 < 0$, so $T$ flips orientation (reflects).

Common types of linear transformations on $\mathbb{R}^2$

• Rotation by $\theta$: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, $\det = 1$.
• Scaling by $k$ in both directions: $kI$, $\det = k^2$.
• Reflection across the x-axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, $\det = -1$.
• Shear along x by factor $k$: $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$, $\det = 1$ (preserves area).
• Projection onto x-axis: $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $\det = 0$ (collapses to a line).

Properties that linear transformations preserve

• The origin: $T(\mathbf{0}) = \mathbf{0}$.
• Straight lines: lines go to lines (possibly degenerate to points).
• The origin being on a line: lines through origin map to lines through origin.
• Parallel lines: parallel stays parallel.
• Ratios of lengths on a line.

Properties they generally don't preserve

• Lengths (unless $T$ is a rotation/reflection).
• Angles (unless $T$ is orthogonal).
• Areas (unless $|\det| = 1$).

Common mistakes

• Assuming $T(x) = mx + b$ is linear. It's only linear if $b = 0$; otherwise it's affine.
• Missing the $T(\mathbf{0}) = \mathbf{0}$ check. It's a quick disqualifier for many candidate maps.
• Confusing functions of one variable with linear maps. $T(x) = x^2$ is not linear.

Try it in the visualization

Drag the two columns of the transformation matrix. A unit square (and a few sample vectors inside) deforms live. Determinant badge shows the area scale factor.