Vector Spaces and Subspaces

Every algebraic fact proved for ${\mathbb{R}}^n$ in the earlier chapters rested on a short list of properties: addition is commutative and associative, scalar multiplication distributes over sums, there is a zero vector, and so on. Those properties are not special to columns of numbers. Polynomials add and scale the same way. So do real-valued functions, matrices of a fixed size, and infinite sequences of measurements. A vector space is the abstraction that isolates exactly those properties, so a theorem proved once from the axioms holds simultaneously in all of these settings.¹

The axioms

The scalars here are real numbers, so $V$ is a real vector space; the same theory holds verbatim with complex scalars, which becomes relevant once complex eigenvalues appear. The two closure axioms, 1 and 6, are the ones that fail most often when a candidate set turns out not to be a vector space, and they are the first thing to check.

From the axioms alone, three facts follow that are used constantly. For every $u$ in $V$ and every scalar $c$,

The first two are statements about the scalar zero and the vector zero; the third identifies the additive inverse promised by Axiom 5 with the scalar multiple $(-1)u$, so subtraction never needs a separate definition.

Examples across five settings

The following sets are all vector spaces under their natural operations, and none is a column of numbers except the first.

• Euclidean space ${\mathbb{R}}^n$. The standard example. Geometric intuition built in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ carries over to every space below.
• The space $\mathbb{S}$ of signals. Doubly infinite sequences $\{y_k\}=(\dots ,y_{-1},y_0,y_1,y_2,\dots )$, added and scaled term by term. A signal is a function defined on the integers, produced whenever a process is sampled at discrete times. These reappear in difference equations.
• Polynomials ${\mathbb{P}}_n$ of degree at most $n$. All expressions $p(t)=a_0+a_{1}t+\cdots +a_n{t}^n$ with real coefficients. Sums and scalar multiples of such polynomials are again polynomials of degree at most $n$, and the zero polynomial is the zero vector.
• Real-valued functions on a set $D$. With $(f+g)(t)=f(t)+g(t)$ and $(cf)(t)=cf(t)$, the set of all functions $D\to \mathbb{R}$ is a vector space. The zero vector is the function identically zero.
• Matrices $M_{m\times n}$. All $m\times n$ real matrices, under entry-wise addition and scalar multiplication.

Once a set is known to be a vector space, every calculation done in ${\mathbb{R}}^n$ transfers to it unchanged. A function $f$ is then a single point in the space of functions, not a graph.

Subspaces

Most vector spaces met in practice arise as a subset of a larger space — the solutions of a system, the polynomials vanishing at a point, the symmetric matrices. A subset that is itself a vector space under the inherited operations is a subspace, and checking it requires only three of the ten axioms rather than all ten.

Properties (a), (b), (c) are Axioms 4, 1, and 6. The remaining axioms — 2, 3, and 7 through 10 — are automatic, because they are identities that already hold for every element of $V$, including those that happen to lie in $H$. Axiom 5 follows from (c) with $c=-1$, since $-u=(-1)u$. So the three conditions guarantee that $H$ is a vector space. Every subspace is a vector space, and every vector space is a subspace of itself; the word subspace signals that a smaller space is being viewed inside a larger one.

Two small subspaces exist inside every vector space $V$: the zero subspace$\{0\}$, containing only the zero vector, and $V$ itself. For polynomials, ${\mathbb{P}}_n$ is a subspace of $\mathbb{P}$, the space of all polynomials, which in turn is a subspace of the space of all functions on $\mathbb{R}$.

When closure fails

A subset can contain the zero vector and still fail to be a subspace, because closure is a condition about all sums or all scalar multiples. A single counterexample settles the matter.

• A line not through the origin in ${\mathbb{R}}^2$ misses $0$ outright, so it is not a subspace. The same holds for any plane in ${\mathbb{R}}^3$ that avoids the origin.
• ${\mathbb{R}}^2$ is not a subspace of ${\mathbb{R}}^3$. Its vectors have two entries, not three, so it is not even a subset of ${\mathbb{R}}^3$. The set of $(s,t,0)$ with $s,t$ real is a genuine subspace that behaves like ${\mathbb{R}}^2$ but is logically distinct from it.
• The first quadrant $\{(x,y):x\ge 0,y\ge 0\}$ contains $0$ and is closed under addition, but scaling by $c=-1$ sends the vector $(1,1)$ to $(-1,-1)$, which leaves the quadrant. Closure under scalar multiplication fails.

Spans are subspaces

The most common way to describe a subspace is as the set of all linear combinations of a fixed collection of vectors. Recall that $\mathrm{Span}\{v_1,\dots ,v_p\}$ is the set of every vector $c_1{v}_1+\cdots +c_p{v}_p$. This set is always a subspace.

The proof checks the three conditions. The zero vector equals $0v_1+\cdots +0v_p$, so it lies in the span. If $u=s_1{v}_1+\cdots +s_p{v}_p$ and $w=t_1{v}_1+\cdots +t_p{v}_p$ are two linear combinations, then

is again a linear combination, and for any scalar $c$,

is one as well. All three conditions hold. The subspace $\mathrm{Span}\{v_1,\dots ,v_p\}$ is called the subspace spanned (or generated) by the set, and a set whose span is $H$ is a spanning set for $H$.

Membership of a specific vector in a span is a consistency question.

Every nonzero subspace of ${\mathbb{R}}^3$ other than ${\mathbb{R}}^3$ itself is either a line through the origin, $\mathrm{Span}\{v\}$ for a nonzero $v$, or a plane through the origin, $\mathrm{Span}\{v_1,v_2\}$ for two vectors that are not multiples. This classification, proved once dimension is available, is worth keeping in mind even for abstract spaces.

Reading a subspace as a span

A subset written with free parameters is recognized as a span by pulling the parameters out as coefficients.

Reading the free parameters off as coefficients and collecting their vectors is the same procedure that produces a spanning set for the null space of a matrix.

The subspace test also applies where no coordinates are available, directly on the defining property.