Affine Subspaces & Affine Mappings

🎯 Learning Objectives

• Define affine subspaces as translates of linear subspaces
• Determine whether a set is an affine subspace
• Parametrise lines and planes in parametric and implicit forms
• Distinguish between linear, affine, and convex combinations
• Identify affine mappings

📋 Prerequisites

• Subspaces (Week 3) — subspaces must contain the origin
• Linear Transformations (Week 5) — linear maps preserve the origin

1. Intuition: Moving Away from the Origin

2. Definition and Examples

2.1 Definition

Definition (Affine Subspace). A subset $A \subseteq \mathbb{R}^n$ is an affine subspace if it can be written as:

where $V$ is a linear subspace of $\mathbb{R}^n$ and $\mathbf{p} \in \mathbb{R}^n$. The dimension of an affine subspace is $\dim(V)$. Example 1: Lines in ℝ²

• Linear subspace: $y = 2x$ (through origin, dimension 1)
• Affine subspace: $y = 2x + 3$ (parallel to $y = 2x$, shifted up by 3)

$y = 2x + 3$ can be written as $(0,3) + \text{span}\{(1,2)\}$. Example 2: Planes in ℝ³

$2x + 3y - z = 5$ is an affine plane. Direction space: $2x + 3y - z = 0$ (through origin) — a linear subspace. Particular point: $(1,1,0)$ works ($2+3-0=5$). So plane = $(1,1,0) + V$ where $V = \{(x,y,z) \mid 2x+3y-z=0\}$.

2.2 Parametric Form

3. Affine Combinations

Definition (Affine Combination). A linear combination $\sum_{i=1}^k c_i \mathbf{v}_i$ is an affine combination if $\sum_{i=1}^k c_i = 1$. Contrast:

• Linear combination: No restriction on coefficients
• Affine combination: Coefficients sum to 1
• Convex combination: Coefficients sum to 1 and $c_i \geq 0$

Example 3: Affine combinations

$(1,2) = \frac{1}{3}(3,0) + \frac{2}{3}(0,3)$ is an affine combination ($\frac{1}{3}+\frac{2}{3}=1$).

It is also a convex combination (coefficients are non-negative).

$(1,2) = 2(3,0) + (-1)(0,3)$ is not an affine combination ($2 + (-1) = 1$, so it is! Wait — $2 + (-1) = 1$, so it IS an affine combination. But it's not convex since $-1 < 0$.) Theorem: A set $A$ is an affine subspace iff it is closed under affine combinations.

4. Affine Mappings

Definition (Affine Mapping). A function $F: \mathbb{R}^n \to \mathbb{R}^m$ is affine if:

where $A$ is an $m \times n$ matrix and $\mathbf{b} \in \mathbb{R}^m$. Every affine map is a linear transformation followed by a translation. Example 4: Affine vs. linear

$F(x,y) = (2x+3y+1, x-4y-2)$ is affine. $T(x,y) = (2x+3y, x-4y)$ is linear (the linear part). The translation vector is $\mathbf{b} = (1, -2)$. Properties:

• $F(\mathbf{0}) \neq \mathbf{0}$ generally (unlike linear maps)
• $F$ preserves affine combinations: $F(\sum c_i \mathbf{v}_i) = \sum c_i F(\mathbf{v}_i)$ when $\sum c_i = 1$
• Affine maps map affine subspaces to affine subspaces

5. Edge Cases & Gotchas

6. Common Pitfalls

❌ Pitfall 1: Thinking affine subspaces are linear subspaces

❌ Pitfall 2: Confusing affine and convex combinations

7. Formula Summary Table

8. 📝 Practice Questions

Q1: Affine line in ℝ²

Parametrise the line $y = 3x + 1$ as an affine subspace.

Solution: Direction vector: $(1,3)$. Point on line: $(0,1)$. $\mathbf{x}(t) = (0,1) + t(1,3) = (t, 1+3t)$. Q2: Is it affine?

Is $A = \{(x,y,z) \mid x + y + z = 2\}$ an affine subspace?

Solution: Direction space $V = \{(x,y,z) \mid x+y+z = 0\}$ (a plane through origin). Point: $(2,0,0) \in A$. So $A = (2,0,0) + V$. Yes, it's affine. Q3: Affine combination

Express $(3,4)$ as an affine combination of $(1,2)$ and $(5,6)$.

Solution: $(3,4) = c(1,2) + (1-c)(5,6)$. $c + 5(1-c) = 3 \Rightarrow c + 5 - 5c = 3 \Rightarrow -4c = -2 \Rightarrow c = 1/2$. Check: $2c + 6(1-c) = 2(1/2) + 6(1/2) = 1+3 = 4$ ✓. So $(3,4) = \frac{1}{2}(1,2) + \frac{1}{2}(5,6)$. Q4: Affine mapping

$F(x,y) = (x+2y+1, 3x-4y+2)$. Find $F(0,0)$ and $F(1,1)$.

Solution: $F(0,0) = (1,2)$. $F(1,1) = (1+2+1, 3-4+2) = (4,1)$. Q5: Affine image of a line

Find the image of the line $\mathbf{x}(t) = (t, 2t)$ under $F(x,y) = (x+y+1, x-2y)$.

Solution: $F(t,2t) = (t+2t+1, t-4t) = (3t+1, -3t)$. This is $\mathbf{y}(t) = (1,0) + t(3,-3) = (1,0) + 3t(1,-1)$, an affine line through $(1,0)$ with direction $(1,-1)$. Q6: Is it affine?

Is $A = \{(x,y) \mid x^2 + y^2 = 1\}$ affine?

Solution: No. The circle is not an affine subspace (it's not a translate of a linear subspace; it fails closure under affine combinations). Q7: Dimension of affine subspace

What's the dimension of the affine plane $2x - y + 3z = 4$?

Solution: The direction space is $2x - y + 3z = 0$, which is a 2-dimensional subspace. So the affine plane has dimension 2. Q8: Affine combination in ℝ³

Express $(2,3,4)$ as an affine combination of $(1,0,1)$, $(0,1,0)$, and $(0,0,1)$.

Solution: Need $c_1 + c_2 + c_3 = 1$ and: $c_1(1,0,1) + c_2(0,1,0) + c_3(0,0,1) = (c_1, c_2, c_1 + c_3) = (2,3,4)$. $c_1 = 2$, $c_2 = 3$, $c_1 + c_3 = 4 \Rightarrow c_3 = 2$. Check sum: $2+3+2 = 7 \neq 1$. Not an affine combination. Q9: Affine vs linear mapping

Is $F(x) = 3x + 1$ (from $\mathbb{R}$ to $\mathbb{R}$) linear?

Solution: No. $F(0) = 1 \neq 0$ and $F(x+y) = 3(x+y)+1 = 3x+3y+1 \neq F(x)+F(y) = 3x+1+3y+1 = 3x+3y+2$. It is affine. Q10: Intersection of affine subspaces

Find the intersection of the affine lines: $L_1: (t, 2t+1)$ and $L_2: (s+1, s+3)$.

Solution: Set $(t, 2t+1) = (s+1, s+3)$: $t = s+1$, $2t+1 = s+3$. Substitute: $2(s+1)+1 = s+3 \Rightarrow 2s+2+1 = s+3 \Rightarrow s = 0$, $t = 1$. Intersection point: $(1, 3)$.

🔗 Cross-References

• Next topic: Inner Products & Norms
• BSMA1001 (Maths 1): Equation of a line $y = mx + c$
• Week 10 (Tangent Planes): Tangent planes are affine subspaces Join Discord Previous7.1 Equivalent & Similar MatricesNext7.3 Inner Products & Norms