Subspaces: Definition, Testing & Examples

🎯 Learning Objectives

• State the subspace test (closed under $+$, closed under $\cdot$, contains $\mathbf{0}$)
• Determine whether a given subset of a vector space is a subspace
• Visualise subspaces of $\mathbb{R}^2$ and $\mathbb{R}^3$ as lines, planes, and points through the origin
• Describe the column space, null space, and row space of a matrix
• Prove that the intersection of two subspaces is a subspace

📋 Prerequisites

• Vector Spaces Definition (this week) — you need to know the axioms
• Subspaces are vector spaces that live inside bigger vector spaces

1. Intuition: A Smaller Room Inside the House

1. Contain the zero vector — the room must have a floor
2. Be closed under addition — adding two things in the room keeps you in the room
3. Be closed under scalar multiplication — stretching/shrinking something in the room keeps you in the room (Diagram) In $\mathbb{R}^3$, subspaces are:

• The origin $\{\mathbf{0}\}$ (dimension 0)
• Lines through the origin (dimension 1)
• Planes through the origin (dimension 2)
• The whole space $\mathbb{R}^3$ (dimension 3) Crucially: Subspaces must contain the origin. If it doesn't pass through the origin, it's not a subspace.

2. The Subspace Test

Theorem (Subspace Test). A non-empty subset $W$ of a vector space $V$ is a subspace of $V$ if and only if:

1. Zero vector: $\mathbf{0} \in W$
2. Closure under addition: For all $\mathbf{u}, \mathbf{v} \in W$, $\mathbf{u} + \mathbf{v} \in W$
3. Closure under scalar multiplication: For all $\mathbf{v} \in W$ and $c \in \mathbb{R}$, $c\mathbf{v} \in W$ If these three hold, the other seven axioms are automatically inherited from $V$. Why only three checks? Because $W$ inherits commutativity, associativity, etc. from $V$. The only things that could go wrong are: (a) $W$ lacks a zero vector, (b) adding two elements of $W$ leaves $W$, or (c) scaling an element of $W$ leaves $W$.

2.1 Streamlined Test

One-step test: $W$ is a subspace iff $\mathbf{0} \in W$ and for all $\mathbf{u}, \mathbf{v} \in W$, $c \in \mathbb{R}$, we have $\mathbf{u} + c\mathbf{v} \in W$.

3. Examples of Subspaces

3.1 Lines Through the Origin in $\mathbb{R}^2$

Example 1: $W = \{(x, y) \in \mathbb{R}^2 \mid y = 2x\}$ Check:

• $\mathbf{0} = (0,0)$: $0 = 2(0)$ ✓
• Closure under addition: Let $(x_1, 2x_1)$, $(x_2, 2x_2) \in W$. Sum = $(x_1+x_2, 2x_1+2x_2) = (x_1+x_2, 2(x_1+x_2))$ — still satisfies $y = 2x$ ✓
• Closure under scalar multiplication: $c(x, 2x) = (cx, 2cx)$ — still satisfies $y = 2x$ ✓ So $W$ is a 1-dimensional subspace of $\mathbb{R}^2$.

3.2 Planes Through the Origin in $\mathbb{R}^3$

Example 2: $W = \{(x,y,z) \in \mathbb{R}^3 \mid x + y - z = 0\}$ Check:

• $\mathbf{0} = (0,0,0)$: $0 + 0 - 0 = 0$ ✓
• Closure under addition: Let $(x_1,y_1,z_1)$, $(x_2,y_2,z_2) \in W$. Then $x_1+y_1-z_1 = 0$, $x_2+y_2-z_2 = 0$. Sum = $(x_1+x_2, y_1+y_2, z_1+z_2)$. Check: $(x_1+x_2)+(y_1+y_2)-(z_1+z_2) = (x_1+y_1-z_1)+(x_2+y_2-z_2) = 0+0 = 0$ ✓
• Closure under scalar multiplication: $c(x,y,z) = (cx, cy, cz)$. Check: $cx + cy - cz = c(x+y-z) = c \cdot 0 = 0$ ✓ So $W$ is a 2-dimensional subspace of $\mathbb{R}^3$.

3.3 Polynomial Subspaces

Example 3: $W = \{p(x) \in P_2 \mid p(0) = 0\}$ (polynomials of degree $\leq 2$ with zero constant term) Check:

• Zero polynomial: $p(x) = 0$ satisfies $p(0) = 0$ ✓
• Closure under addition: If $p(0)=0$ and $q(0)=0$, then $(p+q)(0) = p(0)+q(0) = 0+0 = 0$ ✓
• Closure under scalar multiplication: $(cp)(0) = c\cdot p(0) = c\cdot 0 = 0$ ✓ So $W$ is a subspace of $P_2$. It consists of polynomials like $a_1 x + a_2 x^2$.

4. Non-Examples

4.1 Line Not Through the Origin

$W = \{(x,y) \in \mathbb{R}^2 \mid y = 2x + 1\}$ Fails zero vector: $(0,0)$ gives $0 = 1$, so $\mathbf{0} \notin W$. Not a subspace. Also fails closure under addition: $(x_1, 2x_1+1) + (x_2, 2x_2+1) = (x_1+x_2, 2(x_1+x_2)+2)$. The sum has $y = 2x + 2$, not $2x+1$.

4.2 First Quadrant

$W = \{(x,y) \in \mathbb{R}^2 \mid x \geq 0, y \geq 0\}$ Contains $\mathbf{0}$ ✓, closed under addition ✓, but fails scalar multiplication: $(-1)(1,1) = (-1,-1) \notin W$. Not a subspace.

5. Important Subspaces Associated with a Matrix

5.1 Null Space (Kernel)

Definition (Null Space). For an $m \times n$ matrix $A$, the null space is:

• $\mathbf{0} \in N(A)$ because $A\mathbf{0} = \mathbf{0}$ ✓
• If $A\mathbf{x} = \mathbf{0}$ and $A\mathbf{y} = \mathbf{0}$, then $A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}$ ✓
• If $A\mathbf{x} = \mathbf{0}$, then $A(c\mathbf{x}) = cA\mathbf{x} = c \cdot \mathbf{0} = \mathbf{0}$ ✓

5.2 Column Space (Range)

Definition (Column Space). For an $m \times n$ matrix $A$, the column space is:

5.3 Row Space

Definition (Row Space). The row space of $A$ is the span of its rows (as vectors in $\mathbb{R}^n$). It equals $C(A^T)$. Example 4: Null space and column space

Null space: Solve $A\mathbf{x} = \mathbf{0}$: $x + 2y = 0$, $2x + 4y = 0$. Solutions: $x = -2y$, $y$ free.

— a line through origin.

Column space: Columns are $(1,2)$ and $(2,4)$. The second is a scalar multiple of the first, so $C(A) = \text{span}\{(1,2)\}$ — also a line through origin.

6. Subspace Operations

6.1 Intersection of Subspaces

Theorem. If $W_1$ and $W_2$ are subspaces of $V$, then $W_1 \cap W_2$ is also a subspace of $V$. Proof: $\mathbf{0} \in W_1$ and $\mathbf{0} \in W_2$, so $\mathbf{0} \in W_1 \cap W_2$. If $\mathbf{u}, \mathbf{v} \in W_1 \cap W_2$, then $\mathbf{u}+\mathbf{v} \in W_1$ and $\mathbf{u}+\mathbf{v} \in W_2$, so $\mathbf{u}+\mathbf{v} \in W_1 \cap W_2$. Similarly for scalar multiplication.

6.2 Sum of Subspaces

Definition (Sum). $W_1 + W_2 = \{\mathbf{w}_1 + \mathbf{w}_2 \mid \mathbf{w}_1 \in W_1, \mathbf{w}_2 \in W_2\}$. The sum of two subspaces is also a subspace. Example 5: Intersection and sum

In $\mathbb{R}^3$, let $W_1 = \{(x,y,0)\}$ (the $xy$-plane) and $W_2 = \{(0,y,z)\}$ (the $yz$-plane).

$W_1 \cap W_2 = \{(0,y,0)\}$ — the $y$-axis (a line). $W_1 + W_2 = \mathbb{R}^3$ (every vector can be written as sum of something in $xy$-plane and something in $yz$-plane).

7. Edge Cases & Gotchas

8. Common Pitfalls

❌ Pitfall 1: Checking only closure

❌ Pitfall 2: Assuming any subset of $\mathbb{R}^n$ is a subspace

❌ Pitfall 3: Confusing null space and column space

9. Formula Summary Table

10. 📝 Practice Questions

Q1: Subspace test — line through origin

Is $W = \{(x,y) \in \mathbb{R}^2 \mid x = -2y\}$ a subspace?

Solution: Yes.

• $\mathbf{0} = (0,0)$: $0 = -2(0)$ ✓
• Addition: $(x_1, -\frac{x_1}{2}) + (x_2, -\frac{x_2}{2}) = (x_1+x_2, -\frac{x_1+x_2}{2})$ ✓
• Scalar: $c(x, -\frac{x}{2}) = (cx, -\frac{cx}{2})$ ✓ Q2: Subspace test — plane in ℝ³

Is $W = \{(x,y,z) \in \mathbb{R}^3 \mid 2x - y + 3z = 0\}$ a subspace?

Solution: Yes (plane through origin).

• $\mathbf{0}$: $2(0) - 0 + 3(0) = 0$ ✓
• Addition: $(x_1+x_2, y_1+y_2, z_1+z_2)$. Check: $2(x_1+x_2)-(y_1+y_2)+3(z_1+z_2) = (2x_1-y_1+3z_1)+(2x_2-y_2+3z_2) = 0+0 = 0$ ✓
• Scalar: $c(x,y,z)$. Check: $2(cx)-cy+3(cz) = c(2x-y+3z) = c\cdot0 = 0$ ✓ Q3: Not a subspace — line not through origin

Is $W = \{(x,y) \in \mathbb{R}^2 \mid y = 3x - 1\}$ a subspace?

Solution: No. $(0,0) \notin W$ because $0 \neq -1$. Fails the zero vector condition. Q4: Not a subspace — discrete set

Is $W = \{(x,y) \in \mathbb{R}^2 \mid x^2 = y^2\}$ a subspace?

Solution: No. $(1,1) \in W$ and $(1,-1) \in W$, but $(1,1)+(1,-1) = (2,0)$. Check: $2^2 \neq 0^2$, so $(2,0) \notin W$. Fails closure under addition. Q5: Polynomial subspace

Is $W = \{p(x) \in P_2 \mid p'(0) = 0\}$ a subspace of $P_2$?

Solution: Yes.

• Zero polynomial $p(x)=0$ has $p'(0)=0$ ✓
• If $p'(0)=0$ and $q'(0)=0$, then $(p+q)'(0) = p'(0)+q'(0) = 0+0 = 0$ ✓
• $(cp)'(0) = c p'(0) = c\cdot0 = 0$ ✓ Q6: Null space

Find a basis for $N(A)$ where

.

Solution: Solve $A\mathbf{x} = \mathbf{0}$:

$x = z$, $y = -2z$, $z$ free.

. Basis: $\{(1,-2,1)\}$. Q7: Column space

Find a basis for $C(A)$ where

.

Solution: Columns are $(1,2)$ and $(2,4)$. Since column 2 = 2(column 1), they are linearly dependent. A basis for $C(A)$ is $\{(1,2)\}$. Q8: Intersection of subspaces

In $\mathbb{R}^3$, $W_1 = \{(x,y,z) \mid x + y = 0\}$, $W_2 = \{(x,y,z) \mid y - z = 0\}$. Find $W_1 \cap W_2$.

Solution: $x + y = 0$ and $y - z = 0$ ⇒ $x = -y$, $z = y$. $W_1 \cap W_2 = \{(-y, y, y) \mid y \in \mathbb{R}\} = \text{span}\{(-1,1,1)\}$, a line. Q9: Union is not a subspace

Show that $W_1 \cup W_2$ is not a subspace where $W_1 = \{(x,0)\}$ and $W_2 = \{(0,y)\}$ in $\mathbb{R}^2$.

Solution: $(1,0) \in W_1 \cup W_2$, $(0,1) \in W_1 \cup W_2$, but $(1,0)+(0,1) = (1,1) \notin W_1$ and $(1,1) \notin W_2$, so $(1,1) \notin W_1 \cup W_2$. Fails closure under addition. Q10: Sum of subspaces

If $W_1 = \{(x,y,0)\}$ and $W_2 = \{(0,0,z)\}$ in $\mathbb{R}^3$, show that $W_1 + W_2 = \mathbb{R}^3$.

Solution: Any $(a,b,c) \in \mathbb{R}^3$ can be written as $(a,b,0) + (0,0,c)$ where $(a,b,0) \in W_1$ and $(0,0,c) \in W_2$. So $W_1 + W_2 = \mathbb{R}^3$.

🔗 Cross-References

• Next topic: Linear Independence
• Week 5 (Rank-Nullity): Relates dimensions of $N(A)$ and $C(A)$
• Week 6 (Kernel & Image): Generalises null space and column space to linear transformations
• Week 7 (Affine Subspaces): Subspaces translated away from origin
• BSCS2004 (ML Foundations): Feature spaces, solution spaces in regression Join Discord Previous3.1 Vector Spaces DefinitionNext3.3 Linear Independence