Lesson 1: Linear Transformations -- Definition and Examples
Estimated time: 40-50 minutes
Learning Objectives
- Define a linear transformation between vector spaces
- Verify whether a given function is a linear transformation
- Identify standard transformations: rotation, reflection, projection, and scaling
- Understand the connection between linear transformations and matrix multiplication
What Is a Linear Transformation?
A linear transformation is a function between vector spaces that preserves the two fundamental operations: vector addition and scalar multiplication. Think of it as a function that "plays nicely" with the structure of the vector space.
Definition: A function T : R^n → R^m is a linear transformation if for all vectors u, v in R^n and all scalars c:
(1) T(u + v) = T(u) + T(v) (preserves addition)
(2) T(cu) = cT(u) (preserves scalar multiplication)
These two conditions can be combined into one: T(cu + dv) = cT(u) + dT(v) for all vectors u, v and scalars c, d. This is called superposition.
Quick Consequence
Every linear transformation maps the zero vector to the zero vector: T(0) = T(0 * v) = 0 * T(v) = 0.
This gives a quick test: if T(0) is not 0, then T is NOT linear.
Verifying Linearity
Worked Example 1: A Linear Transformation
Let T : R^2 → R^2 be defined by T(x, y) = (2x + y, 3x - y).
Check addition: Let u = (x1, y1) and v = (x2, y2).
T(u + v) = T(x1+x2, y1+y2) = (2(x1+x2) + (y1+y2), 3(x1+x2) - (y1+y2))
= (2x1+y1, 3x1-y1) + (2x2+y2, 3x2-y2) = T(u) + T(v). ✓
Check scalar multiplication: T(cu) = T(cx1, cy1) = (2cx1 + cy1, 3cx1 - cy1) = c(2x1+y1, 3x1-y1) = cT(u). ✓
T is linear. In fact, T(x,y) = [2 1; 3 -1] * [x; y] -- it is matrix multiplication.
Worked Example 2: NOT Linear
Let T : R^2 → R^2 be defined by T(x, y) = (x + 1, y).
Quick test: T(0, 0) = (0 + 1, 0) = (1, 0) ≠ (0, 0).
Since T(0) ≠ 0, T is not linear. The "+1" is a translation, which breaks linearity.
Worked Example 3: NOT Linear
Let T : R^2 → R be defined by T(x, y) = x * y.
Check: T(2, 3) = 6. But T(2*(1,1)) = T(2,2) = 4, while 2*T(1,1) = 2*1 = 2. Since 4 ≠ 2, T is not linear.
Products of components are never linear -- they are quadratic.
Standard Transformations in R^2
These are the building blocks of geometry in linear algebra. Each one is a linear transformation described by a 2x2 matrix.
Rotation by angle θ: T(x, y) rotates the vector by θ counterclockwise about the origin.
[sinθ cosθ]
Example: Rotation by 90 degrees
cos(90) = 0, sin(90) = 1. Matrix: [0 -1; 1 0].
T(1, 0) = (0, 1). T(0, 1) = (-1, 0). T(3, 2) = (-2, 3). ✓
Reflection across the x-axis: T(x, y) = (x, -y). Matrix: [1 0; 0 -1].
Reflection across the y-axis: T(x, y) = (-x, y). Matrix: [-1 0; 0 1].
Reflection across y = x: T(x, y) = (y, x). Matrix: [0 1; 1 0].
Projection onto the x-axis: T(x, y) = (x, 0). Matrix: [1 0; 0 0].
Projection onto the y-axis: T(x, y) = (0, y). Matrix: [0 0; 0 1].
Example: Projection
Let T project onto the x-axis: T(x,y) = (x, 0).
T(3, 5) = (3, 0). T(0, 7) = (0, 0). The entire y-component is "lost."
Notice: T(T(v)) = T(v). Applying a projection twice gives the same result as once. This property is called idempotence.
Scaling: T(x, y) = (kx, ky) for scalar k. Matrix: [k 0; 0 k] = kI.
This scales every vector by factor k. If k < 0, it also reverses direction.
Linear Transformations and Matrix Multiplication
Key Fact: Every linear transformation T : R^n → R^m can be written as T(x) = Ax for some m x n matrix A. Conversely, every matrix defines a linear transformation.
This is the central insight of the chapter: linear transformations and matrices are two views of the same thing. We will see how to construct A in Lesson 2.
Check Your Understanding
1. Is T(x, y) = (3x, x + y, 0) a linear transformation from R^2 to R^3?
2. Is T(x, y) = (x^2, y) linear?
3. What matrix performs reflection across the line y = x?
4. If T is linear and T(0) = (5, 2), what can you conclude?
Key Takeaways
- Linearity means T(cu + dv) = cT(u) + dT(v) -- addition and scalar multiplication are preserved
- Quick non-linearity test: If T(0) ≠ 0, then T is not linear
- Standard 2D transformations -- rotation, reflection, projection, scaling -- are all linear
- Translations (adding a constant vector) are NOT linear
- Every linear transformation from R^n to R^m is multiplication by an m x n matrix