Subspaces

Subtopic: Vector Spaces

A subspace is a vector space living inside a larger vector space—a subset that is itself closed under addition and scalar multiplication. Subspaces are everywhere: lines and planes through the origin in (R^3), solution sets of homogeneous equations, and null spaces of matrices. Understanding subspaces is essential for decomposing spaces and characterizing linear maps.

Introduction

Imagine (R^3), the space of all 3D vectors. Some subsets of (R^3) have special structure: a plane through the origin contains all linear combinations of two vectors in it. A line through the origin is closed under scaling and addition (of vectors along that line). These are subspaces.

But not every subset is a subspace. A plane that doesn't pass through the origin isn't closed under scalar multiplication—scaling a vector on that plane might land outside it. A sphere isn't closed under addition—add two points on a sphere and you typically land outside it.

Subspaces capture the "flat" pieces of a vector space that preserve the linear structure.

Definition

Let V be a vector space over a field F. A subset W⊆V is a subspace of V if W is itself a vector space under the same operations.

Equivalently (and more practically), W is a subspace if and only if:

• W is nonempty (typically verified by checking 0∈W)

• W is closed under addition: if u,v∈W, then u+v∈W

• W is closed under scalar multiplication: if v∈W and c∈F, then c*v∈W

These three conditions are called the subspace test. You do not need to check all eight vector space axioms—they are inherited from V.

Geometric Interpretation

In (R^2), the subspaces are: the zero vector alone {0}, lines through the origin, and (R^2) itself.

In (R^3), the subspaces are: {0}, lines through the origin, planes through the origin, and (R^3) itself.

The key geometric insight: subspaces must pass through the origin. A line or plane not through the origin fails because 0 is not in the set, and scaling by 0 must produce the zero vector.

You can visualize a subspace as a “flat slice” through the origin. In higher dimensions, subspaces are hyperplanes (of various dimensions) passing through the origin.

The Subspace Test

To prove W is a subspace, verify these three conditions:

Step 1: Non-empty

Show that at least one element is in W. The easiest way: show 0∈W. Since any subspace must contain the zero vector (0=0⋅v∀v), this is necessary anyway.

Step 2: Closure under addition

Take two arbitrary elements u, v∈W and show that u+v∈W.

Step 3: Closure under scalar multiplication

Take any v∈W and any scalar c, and show that c*v∈W.

If any step fails, W is not a subspace. Provide a counterexample: specific elements that violate closure.

Worked Example

Let W={(x,y,z)∈(R^3):x+2*y−z=0}. Is W a subspace of (R^3)?

Step 1: Check that 0∈W

Substitute (0,0,0): 0+2*(0)−0=0.

So the zero vector is in W.

Step 2: Check closure under addition

Let u=((x_1),(y_1),(z_1)) and v=((x_2),(y_2),(z_2)) be in W. Then:

(x_1)+2*(y_1)−(z_1)=0
(x_2)+2*(y_2)−(z_2)=0

We check whether u+v=((x_1)+(x_2),(y_1)+(y_2),(z_1)+(z_2))
satisfies the defining equation:((x_1)+(x_2))+2*((y_1)+(y_2))−((z_1)+(z_2))=((x_1)+2*(y_1)−(z_1))+((x_2)+2*(y_2)−(z_2))=0+0=0

Step 3: Check closure under scalar multiplication

Let v=(x,y,z)∈W and let c be any scalar. Then x+2*y−z=0.

Check whether c*v=(c*x,c*y,c*z) is in W: c*x+2*(c*y)−c*z=c*(x+2*y−z)=c(0)=0

All three conditions hold, so W is a subspace of (R^3).

Geometrically, W is a plane through the origin (solutions to a homogeneous linear equation).

Non-Example

Let W={(x,y)∈R2:x+y=1}. Is this a subspace?

Check if 0∈W: 0+0=0≠1. The zero vector is not in W.

W is NOT a subspace. Geometrically, it is a line that does not pass through the origin.

Important Subspaces

Null Space (Kernel)

For a matrix A, the null space is Null(A)={x:A*x=0}. This is always a subspace (the solution set of a homogeneous system).

Column Space (Range)

For a matrix A, the column space is Col(A)={A*x:x∈Rn} — all possible outputs of the linear map. This is a subspace of the codomain.

Span

Given vectors (v_1),…,(v_k), their span is all linear combinations (c_1)*(v_1)+⋯+(c_k)*(v_k).
The span is always a subspace — the smallest subspace containing those vectors.

Properties of Subspaces

• Every vector space V has at least two subspaces: {0} (the trivial subspace) and V itself.

• The intersection of subspaces is a subspace. If (W_1) and (W_2) are subspaces of V, then (W_1)∩(W_2) is also a subspace.

• The union of subspaces is generally NOT a subspace (unless one contains the other).

• The sum of subspaces (W_1)+(W_2)={(w_1)+(w_2):(w_1)∈(W_1),(w_2)∈(W_2)} is a subspace.

Applications and Connections

In solving linear systems A*x=b, the solution set is either empty (no solution), a single point (unique solution), or a subspace translated by a particular solution (infinitely many solutions of the form x=(x_p)+Null(A)).

In machine learning, PCA finds subspaces that capture the most variance in data.

In quantum mechanics, possible states of a system form a subspace of the full Hilbert space, constrained by physical conditions.

The dimension of a subspace becomes crucial for the rank–nullity theorem and for understanding linear transformations.

Summary

A subspace is a subset of a vector space that is itself a vector space under the same operations. To verify: check that 0 is in the set, and the set is closed under addition and scalar multiplication. Key examples include null spaces, column spaces, and spans. Subspaces must pass through the origin—sets defined by non-homogeneous equations, like x+y=1, are not subspaces.