Vector subspaces

Most of the time we do not build a vector space from scratch: we isolate a piece of one inside another. That piece is called a vector subspace.

The test to apply

A subset F of a vector space E is a vector subspace if:

1. F contains the zero vector           0 ∈ F
2. F is closed under addition           u, v ∈ F  ->  u + v ∈ F
3. F is closed under scalar multiples   u ∈ F, λ real  ->  λu ∈ F

Points 2 and 3 merge into the single test used in practice:

   0 ∈ F     and     for all u, v ∈ F and all λ, μ :   λu + μv ∈ F

The gain is considerable: checking these three points is far quicker than re-checking the eight axioms. A vector subspace is automatically a vector space.

The zero-vector test

It eliminates a large share of candidates in one second. If 0 is not in it, it is not a vector subspace:

F = { (x ; y) with 2x - y = 0 }     0 = (0;0) satisfies 0 - 0 = 0   ✔ subspace
G = { (x ; y) with 2x - y = 3 }     0 does not satisfy 0 - 0 = 3    ✘ not one

Geometrically, F is a line through the origin, G a parallel line that misses O.

What subspaces look like

In R³ the list is short and complete:

   {0}              a line             a plane            all of R³
                    through O          through O
    •                    /             ________
                        /  •          /_______/
                       /             (through the origin)

dimension 0        dimension 1        dimension 2       dimension 3

Every subspace contains the origin and extends indefinitely in its directions. A segment, a disc, a sphere or a parabola is therefore never a vector subspace.

The two ways to build one

By equations (implicit description): the solution set of a homogeneous system is always a vector subspace.

F = { (x;y;z) : x + 2y - z = 0 }        a plane in R³

By generators (parametric description): take some vectors and form all their combinations. The result is written Vect(u, v) and called the spanned subspace.

Vect(u) = all multiples λu            -> a line
Vect(u, v) = { λu + μv }              -> a plane (if u and v are not collinear)

Going from one description to the other — equations ⟷ generators — is exactly what Gaussian elimination does.

Intersection and sum

F ∩ G   (intersection)  ->  ALWAYS a vector subspace
F ∪ G   (union)         ->  almost never
F + G = { u + v }       ->  always a vector subspace

The union of two distinct lines in the plane is not closed: adding a vector from one to a vector from the other takes you off both lines. The sum F + G is defined precisely to repair this.

Summary

• A vector subspace contains 0 and is closed under linear combination.
• The zero-vector test immediately rules out candidates shifted away from the origin.
• In R³: {0}, a line through O, a plane through O, or all of R³.
• The solutions of a homogeneous system form a vector subspace.
• Vect(u, v) — the set of combinations — is the spanned subspace.
• The intersection is a subspace, the union almost never is.