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Linear
Transformations

Understand what makes a function between vector spaces linear: the two properties that preserve addition and scalar multiplication.

Linear Algebra

What Is a Transformation?

A transformation is simply a function that takes a vector as input and produces a vector as output. We write \( T : V \to W \), meaning \( T \) maps vectors from vector space \( V \) to vector space \( W \).

V (Domain) v T W (Codomain) T(v)

Why Does "Linear" Matter?

Not every transformation is linear. A linear transformation preserves the fundamental structure of vector spaces: it respects how vectors add together and how they scale. This makes linear transformations predictable, well-behaved, and incredibly powerful.

Predictability
Knowing \( T \) on basis vectors tells you \( T \) on every vector.
Matrix Representation
Every linear transformation can be represented as matrix multiplication.
Applications
Computer graphics, data science, physics, and engineering all rely on linearity.

The Two Key Properties

A transformation \( T : V \to W \) is linear if and only if it satisfies both of these properties for all vectors \( \mathbf{u}, \mathbf{v} \in V \) and all scalars \( c \in \mathbb{R} \):

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

Formal Definition

V u v u+v T T T W T(u) T(v) T(u)+T(v) T(u+v) must equal T(u)+T(v) Additivity preserved!

Visualizing Additivity

Input Space O
\( \mathbf{u} \)
\( \mathbf{v} \)
\( \mathbf{u}+\mathbf{v} \)
\( T \)
 Output Space O
\( T(\mathbf{u}) \)
\( T(\mathbf{v}) \)
\( T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v}) \)

Visualizing Homogeneity

Input Space
\( \mathbf{v} \)
\( 2\mathbf{v} \)
\( T \)
 Output Space
\( T(\mathbf{v}) \)
\( T(2\mathbf{v})=2T(\mathbf{v}) \)
 Scaling by c=2 is preserved through T

Linear vs Nonlinear

Linear Nonlinear
Additivity ✓ Always holds ✕ May fail
Homogeneity ✓ Always holds ✕ May fail
\( T(\mathbf{0}) \) \( \mathbf{0} \) always Could be anything
Matrix form \( T(\mathbf{x}) = A\mathbf{x} \) No matrix form
Example \( T(x,y) = (2x, x+y) \) \( T(x,y) = (x^2, y+1) \)

Consequence: \( T(\mathbf{0}) = \mathbf{0} \)

If \( T \) is linear, then it must map the zero vector to the zero vector. Here's why:

1
Pick any \( \mathbf{v} \in V \). We know \( \mathbf{0} = \mathbf{v} - \mathbf{v} = \mathbf{v} + (-1)\mathbf{v} \).
2
By additivity: \( T(\mathbf{0}) = T(\mathbf{v} + (-1)\mathbf{v}) = T(\mathbf{v}) + T((-1)\mathbf{v}) \).
3
By homogeneity: \( T((-1)\mathbf{v}) = (-1)T(\mathbf{v}) = -T(\mathbf{v}) \).
4
Therefore: \( T(\mathbf{0}) = T(\mathbf{v}) + (-T(\mathbf{v})) = \mathbf{0} \). ✓

How to Check Linearity

1
Quick check: Compute \( T(\mathbf{0}) \). If \( T(\mathbf{0}) \neq \mathbf{0} \), stop: it is not linear.
2
Test additivity: Pick generic \( \mathbf{u}, \mathbf{v} \). Compute both \( T(\mathbf{u}+\mathbf{v}) \) and \( T(\mathbf{u}) + T(\mathbf{v}) \). Are they equal?
3
Test homogeneity: Pick generic \( c, \mathbf{v} \). Compute both \( T(c\mathbf{v}) \) and \( cT(\mathbf{v}) \). Are they equal?
4
Or combined: Verify \( T(c_1\mathbf{u} + c_2\mathbf{v}) = c_1 T(\mathbf{u}) + c_2 T(\mathbf{v}) \) for all \( c_1, c_2, \mathbf{u}, \mathbf{v} \).

Example 1: Proving Linearity

Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( T(x, y) = (2x + y, \; 3y) \). Is \( T \) linear?

1
Let \( \mathbf{u} = (x_1, y_1) \) and \( \mathbf{v} = (x_2, y_2) \).
2
Additivity: \( T(\mathbf{u}+\mathbf{v}) = T(x_1+x_2,\; y_1+y_2) = (2(x_1+x_2)+(y_1+y_2),\; 3(y_1+y_2)) \).
3
\( T(\mathbf{u})+T(\mathbf{v}) = (2x_1+y_1, 3y_1)+(2x_2+y_2, 3y_2) = (2x_1+2x_2+y_1+y_2,\; 3y_1+3y_2) \). ✓ Equal!
4
Homogeneity: \( T(c\mathbf{v}) = T(cx_2, cy_2) = (2cx_2+cy_2, 3cy_2) = c(2x_2+y_2, 3y_2) = cT(\mathbf{v}) \). ✓ Equal!

Example 2: Disproving Linearity

Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( T(x, y) = (x^2, \; y) \). Is \( T \) linear?

1
Quick check: \( T(0, 0) = (0^2, 0) = (0, 0) \). Passes, but not conclusive.
2
Test homogeneity with \( c = 2 \) and \( \mathbf{v} = (1, 0) \).
3
\( T(2 \cdot (1,0)) = T(2,0) = (4, 0) \).
4
\( 2 \cdot T(1,0) = 2 \cdot (1,0) = (2, 0) \).
5
\( (4,0) \neq (2,0) \). ✕ Homogeneity fails!

Example 3: The Translation Trap

Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( T(x, y) = (x + 1, \; y) \). Is \( T \) linear?

1
Quick check: \( T(0, 0) = (0 + 1, 0) = (1, 0) \neq (0, 0) \).
2
✕ Already fails! The zero vector is not mapped to zero.

Example 4: Between Different Spaces

Let \( T : \mathbb{R}^3 \to \mathbb{R}^2 \) be defined by \( T(x, y, z) = (x - 2z, \; y + z) \). Show \( T \) is linear.

Common Mistakes

✕ Forgetting Constants
Assuming \( T(x,y) = (x+1, y) \) is linear because it "looks simple." Any additive constant kills linearity.
✕ Testing Only Numbers
Checking \( T \) on specific vectors like \( (1,0) \) does not prove linearity. You must prove it for all vectors.
✕ Confusing Linear Functions
In calculus, \( f(x) = 3x + 2 \) is "linear." In linear algebra, it is not: \( f(0) = 2 \neq 0 \).

Summary of Linear Transformations

Property Details
Notation \( T : V \to W \)
Additivity \( T(\mathbf{u}+\mathbf{v}) = T(\mathbf{u})+T(\mathbf{v}) \)
Homogeneity \( T(c\mathbf{v}) = cT(\mathbf{v}) \)
Combined (Superposition) \( T(c_1\mathbf{u}+c_2\mathbf{v}) = c_1T(\mathbf{u})+c_2T(\mathbf{v}) \)
Zero vector \( T(\mathbf{0}) = \mathbf{0} \) always
Matrix form \( T(\mathbf{x}) = A\mathbf{x} \) for some matrix \( A \)
Counterexample method One failure disproves linearity

Question 1 of 5

Which of the following is a requirement for a transformation \( T : V \to W \) to be linear?

Question 2 of 5

Let \( T(x,y) = (x + y, \; 2x - y, \; 0) \). What can you conclude about \( T \)?

Question 3 of 5

Which transformation is not linear?

Question 4 of 5

If \( T \) is a linear transformation and \( T(\mathbf{e}_1) = (2, -1) \), \( T(\mathbf{e}_2) = (0, 5) \), what is \( T(3, 4) \)?

Question 5 of 5

The superposition property \( T(c_1\mathbf{u} + c_2\mathbf{v}) = c_1 T(\mathbf{u}) + c_2 T(\mathbf{v}) \) combines which two properties?