Vector Spaces and Subspaces

A vector space is a set with an addition and a scaling that obey fixed rules. A subspace is a subset that is a vector space in its own right, which three checks decide: it contains the zero vector, and it is closed under addition and under scaling. The checks are separable, and a set can pass one while failing another.

Definition

A vector space over a field $F$ is a set $V$ with two operations, addition $V \times V \to V$ and scalar multiplication $F \times V \to V$ , satisfying, for all $u, v, w \in V$ and $a, b \in F$ :

	Axiom
associativity	$(u + v) + w = u + (v + w)$
commutativity	$u + v = v + u$
additive identity	there is $0 \in V$ with $v + 0 = v$
additive inverse	for each $v$ there is $- v$ with $v + (- v) = 0$
scalar identity	$1 v = v$
scalar associativity	$a (b v) = (a b) v$
distributivity over vectors	$a (u + v) = a u + a v$
distributivity over scalars	$(a + b) v = a v + b v$

The elements are called vectors whatever they are. $R^{n}$ is the familiar example; so are the $m \times n$ matrices, the polynomials of degree at most $n$ , and the real-valued functions on an interval, each with its natural addition and scaling.

Subspaces. A subset $W \subseteq V$ is a subspace when it is itself a vector space under the operations inherited from $V$ . Rechecking all eight axioms is unnecessary: associativity, commutativity and the distributive laws are inherited, because they are statements about elements of $V$ that remain true of elements of $W$ . What can fail is whether the operations stay inside $W$ , so three conditions decide it:

$0 \in W$ ;
$u, v \in W ⟹ u + v \in W$ (closure under addition);
$v \in W$ and $a \in F ⟹ a v \in W$ (closure under scaling).

Given condition 3 and $W$ nonempty, condition 1 follows by taking $a = 0$ . Checking for the zero vector first is nonetheless the cheapest test available, and it disposes of every set defined by an inhomogeneous equation.

Span. For $S = {v_{1}, \dots, v_{k}} \subseteq V$ ,

span (S) = {a_{1} v_{1} + \dots + a_{k} v_{k} : a_{i} \in F},

the set of all linear combinations of $S$ . It is always a subspace, a sum of combinations is a combination, and a scalar times a combination is a combination, and it is the smallest subspace containing $S$ .

Assumptions and scope

The field matters. A set closed under real scaling need not be closed under complex scaling, and a space over $C$ viewed as a space over $R$ has a different dimension.
A subspace inherits its operations. A subset with a different addition defined on it is not a subspace of the original space, however similar the underlying set looks.
The three subspace conditions are not independent: nonemptiness plus closure under scaling gives the zero vector. Checking for zero first is a practical shortcut rather than a logically required step.
A single counterexample settles a closure question, but no finite number of examples establishes closure. Passing must be argued for arbitrary elements.
The span of the empty set is ${0}$ , not the empty set. A vector space always has at least one element.

Worked material

Non-example

Four flawed subspace arguments

Checking examples instead of arbitrary elements. "Take $(1, 0, 1)$ and $(0, 1, 2)$ in $W_{1}$ ; their sum is $(1, 1, 3)$ , which satisfies the equation. So $W_{1}$ is closed under addition." Two members summing correctly is consistent with closure and does not establish it, $W_{3}$ above passes many such checks. Closure is a claim about every pair, so the argument must be carried out with symbols.

Concluding from one closure to the other. "Every multiple of a member stays in the set, so the set is closed." $W_{3}$ is the counterexample: closed under scaling, not under addition. The two conditions constrain different things and neither implies the other.

Treating the zero test as sufficient. "The origin is in the set, so it is a subspace." The origin lies in $W_{3}$ , which is not one. The zero test is a fast way to rule sets out; it rules nothing in.

Mistaking a shifted flat for a subspace. " $x + 2 y - z = 5$ describes a plane, and planes are subspaces." Only planes through the origin are. This one is parallel to $W_{1}$ and misses the origin, and it is a coset of a subspace rather than a subspace. The solution set of an inhomogeneous system, where the homogeneous version gives the subspace.

The common thread: each treats a necessary condition, or a verified instance, as though it settled a universal claim. The subspace test is three separate universal statements, and a verdict of yes requires all three argued in general, while a verdict of no requires only one explicit witness.

Example

Spaces that are not $R^{n}$

The definition earns its generality only if it is used away from coordinates. Four spaces where the same three conditions apply unchanged.

Matrices. $M_{2 \times 2} (R)$ under entrywise addition and scaling. The symmetric matrices form a subspace: the zero matrix is symmetric, a sum of symmetric matrices is symmetric, and a multiple of one is symmetric. The invertible matrices do not. The zero matrix is not invertible, and $(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) + (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix})$ is invertible while neither summand is, so closure fails in both directions at once.

Polynomials. $P_{3}$ , the polynomials of degree at most 3. Those with $p (1) = 0$ form a subspace: the zero polynomial vanishes at 1, and sums and multiples of polynomials vanishing at 1 vanish at 1. Those of degree exactly 3 do not. The zero polynomial is excluded, and $(x^{3} + 1) + (- x^{3}) = 1$ has degree 0.

Functions. Real-valued functions on $[0, 1]$ . The continuous ones form a subspace, since sums and multiples of continuous functions are continuous. So do the differentiable ones, and the solutions of $f^{″} + f = 0$ . The last being why linear differential equations have solution spaces whose members are combinations of a few basis solutions.

Sequences. Infinite real sequences, with the eventually-zero ones forming a subspace of them.

What the pattern shows. In every case the subspaces are the sets defined by conditions that are linear and homogeneous, vanishing at a point, equalling one's own transpose, satisfying a homogeneous differential equation. Conditions that are inhomogeneous ( $p (1) = 2$ ), or nonlinear (degree exactly 3, invertibility, $x y = 0$ ), fail. That is the same distinction as the $R^{3}$ case, stated without coordinates.

Common errors

Common misconception

A set containing the zero vector and closed under scalar multiplication is a subspace, because closure under addition follows from being able to scale members.

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Vector Spaces and Subspaces

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Introduction to Linear Algebra (2016)

Vector Spaces and Subspaces

Assumptions and scope

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Four flawed subspace arguments

Spaces that are not $R^{n}$

Common errors

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Spaces that are not $R^{n}$