Linear Transformations and Matrices

Chapter 2. Linear Transformations and Matrices

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Chapter 2

2.1 Linear Transformations, Null Spaces, and Range

2.2 Matrix Representation

2.3 Composition & Multiplication

2.4 Invertibility & Isomorphisms

2.5 Change of Coordinate Matrix

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1. Linear Transformations, Null Spaces, and Range

Definition — Linear Transformation

A function T: V → W between vector spaces is a linear transformation if for all u, v ∈ V and scalar c:

T(u + v) = T(u) + T(v)     (additivity)
T(c·v)   = c·T(v)          (homogeneity)

Consequence: T(0) = 0 and T(au + bv) = aT(u) + bT(v) for all scalars a, b.

Definition — Null Space & Range

Term	Symbol	Definition
Null Space (Kernel)	N(T)	{ v ∈ V : T(v) = 0 }
Range (Image)	R(T)	{ T(v) : v ∈ V }

Both N(T) and R(T) are subspaces of V and W respectively.

Theorem — Rank–Nullity (Dimension Theorem)

For a linear map T: V → W where V is finite-dimensional:

nullity(T) + rank(T) = dim(V)

i.e.  dim(N(T)) + dim(R(T)) = n

Visual: How T Acts on the Plane (R² → R²)

Standard basis vectors          After transformation T
                                  (example: shear matrix)

  y                               y
  │  e₂=(0,1)                     │    T(e₂)=(1,1)
  │  ↑                            │    ↗
  │  │                            │   /
  │  └──→ e₁=(1,0)               │  /
  └─────────── x                  │ └──→ T(e₁)=(1,0)
                                  └──────────── x

  Unit square □                   Parallelogram ▱
  corners: (0,0)(1,0)(1,1)(0,1)   corners: (0,0)(1,0)(2,1)(1,1)

Common Examples

Transformation	Matrix A	Effect
Identity	[[1,0],[0,1]]	No change
Scale by k	[[k,0],[0,k]]	Uniform scaling
Rotation by θ	[[cosθ,-sinθ],[sinθ,cosθ]]	Rotates by angle θ
Horizontal shear	[[1,k],[0,1]]	Skews horizontally
Reflection (x-axis)	[[1,0],[0,-1]]	Flips over x-axis
Projection (x-axis)	[[1,0],[0,0]]	Collapses to x-axis

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2. The Matrix Representation of a Linear Transformation

Setup

Let V and W be finite-dimensional vector spaces with ordered bases:

• β = {v₁, …, vₙ} for V
• γ = {w₁, …, wₘ} for W

The matrix representation [T]ᵝᵞ is the m × n matrix whose j-th column is the coordinate vector [T(vⱼ)]ᵞ.

         ┌                    ┐
         │ ↑     ↑       ↑   │
[T]ᵝᵞ =  │ [T(v₁)]ᵞ  …  [T(vₙ)]ᵞ │
         │ ↓     ↓       ↓   │
         └                    ┘

Key Fact

Once bases are chosen, every linear transformation T: ℝⁿ → ℝᵐ is uniquely determined by an m × n matrix A, and:

T(x) = Ax   for all x ∈ ℝⁿ

How to Find the Matrix

Step 1. Apply T to each basis vector: compute T(v₁), T(v₂), …, T(vₙ)

Step 2. Express each result in terms of the output basis γ

Step 3. Write those coordinate vectors as columns of A

Example: T: ℝ² → ℝ² where T(x,y) = (2x+y, x−y)

T(e₁) = T(1,0) = (2, 1)   ← column 1
T(e₂) = T(0,1) = (1,−1)   ← column 2

        ┌ 2   1 ┐
A  =    │       │
        └ 1  −1 ┘

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3. Composition of Linear Transformations and Matrix Multiplication

Why Matrix Multiplication Works the Way It Does

If T: V → W and U: W → X are both linear, then the composition UT: V → X is also linear.

In matrix form:

[UT]ᵝᵟ  =  [U]ᵞᵟ · [T]ᵝᵞ

This is precisely why matrix multiplication is defined as it is — it encodes function composition.

Composition in Steps

         T              U
  ℝⁿ ─────→ ℝᵐ ─────→ ℝᵖ
  │                      ↑
  └─────── UT ───────────┘

  x  →  T(x)  →  U(T(x))
  x  →   Ax   →    B(Ax) = (BA)x

Properties

Property	Formula
Associativity	(AB)C = A(BC)
Distributivity	A(B+C) = AB + AC
NOT commutative	AB ≠ BA in general
Identity	AI = IA = A

Example: Rotation then Scaling

T₁ = rotate 45°        T₂ = scale x by 2

     [cos45  -sin45]        [2  0]
A₁ = [sin45   cos45]   A₂= [0  1]

Composed matrix A₂A₁:
  [2·cos45  -2·sin45]   ≈  [ 1.41  -1.41]
  [  sin45    cos45 ]      [ 0.71   0.71]

Note: Apply A₁ FIRST (rightmost), then A₂

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4. Invertibility and Isomorphisms

Definitions

A linear transformation T: V → W is invertible if there exists T⁻¹: W → V such that:

T⁻¹ ∘ T = Iᵥ    (identity on V)
T ∘ T⁻¹ = I_W   (identity on W)

An invertible linear map is called an isomorphism. If such a map exists, V and W are isomorphic (V ≅ W).

Theorem — Invertibility Conditions

For a linear map T: V → W with dim(V) = dim(W) = n, these are equivalent:

1. T is invertible
2. T is injective (one-to-one): N(T) = {0}
3. T is surjective (onto): R(T) = W
4. rank(T) = n
5. det(A) ≠ 0 (for the matrix representation A)

The Determinant as a Scaling Factor

det(A) tells you how A transforms AREA (in 2D) or VOLUME (in 3D)

det > 0  →  orientation preserved, area scaled by |det|
det < 0  →  orientation reversed
det = 0  →  space collapses! NOT invertible

Visual: Effect of det on Unit Square

det = 1          det = 2          det = 0
(no change)      (area doubled)   (collapsed!)

┌──┐             ┌────┐           /
│  │    →        │    │   →      /
└──┘             └────┘         /  ← just a line
area = 1         area = 2       area = 0

Computing the 2×2 Inverse

    ┌ a  b ┐              1    ┌  d  -b ┐
A = │      │   A⁻¹ =  ─────── │       │
    └ c  d ┘            ad-bc  └ -c   a ┘

Valid only when det(A) = ad - bc ≠ 0

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5. The Change of Coordinate Matrix

Motivation

The same linear transformation can have different matrix representations depending on which basis you choose. A smart basis choice can reveal structure (e.g., make A diagonal).

Definition

If β and β’ are two ordered bases for V, the change of coordinate matrix from β’ to β is the invertible matrix Q such that:

[v]_β  =  Q · [v]_β'    for all v ∈ V

Relationship Between Representations

[T]_β'  =  Q⁻¹ · [T]_β · Q

Two matrices related by this formula are called similar.

How to Compute Q

Let β = {u₁, u₂} and β’ = {w₁, w₂}.

Step 1. Express each β'-basis vector in terms of β:
        write w₁ and w₂ as linear combinations of u₁, u₂

Step 2. The coordinate vectors [w₁]_β and [w₂]_β become
        the COLUMNS of Q

Step 3. Verify: Q is invertible (det ≠ 0)

Example: Standard vs. Rotated Basis

Standard basis β:   e₁ = (1,0),  e₂ = (0,1)

Rotated basis β':   w₁ = (cos30°, sin30°)  ≈  (0.87, 0.50)
                    w₂ = (−sin30°, cos30°) ≈  (−0.50, 0.87)

       ┌ 0.87  −0.50 ┐
Q  =   │             │
       └ 0.50   0.87 ┘

A vector v = (1, 1):
  [v]_β  = (1, 1)   (standard coordinates)
  [v]_β' = Q⁻¹(1,1) ≈ (1.37, 0.37)   (rotated coordinates)

Key Insight

Choosing the eigenvectors of T as a basis makes [T]_β diagonal — the simplest possible representation. This is the core motivation for eigenvalue decomposition (Chapter 5).

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Summary

Concept	Key Formula	Condition
Linear map	T(au+bv) = aT(u)+bT(v)	Always
Rank–Nullity	rank(T) + nullity(T) = dim(V)	Finite dim
Matrix of T	Columns = [T(vⱼ)]_γ	Bases β, γ fixed
Composition	[UT] = [U]·[T]	Matching spaces
Invertibility	det(A) ≠ 0 ↔ N(T) = {0} ↔ R(T) = W	Square matrix
Change of basis	[T]_β' = Q⁻¹[T]_βQ	Same V, diff bases

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