Examples of Vector Spaces

What do arrows in the plane, polynomials, matrices, and infinite sequences have in common? At first glance, nothing. One is geometric, one is algebraic, one is tabular, one is combinatorial. Yet all four obey exactly the same ten rules. These rules are vector addition and scalar multiplication. Together they define a vector space. The power of the vector space axioms is precisely their abstractness. They unify these wildly different objects under one theory. A single proof about, say, linear independence applies simultaneously to geometric vectors, polynomials, and matrices. This article is a gallery. Rather than proving new general theorems, its job is different. It builds the mental library of examples every linear algebra student needs. These are the spaces you will test conjectures against. They are the spaces that appear in every later theorem as "let VV be a vector space." They are also the spaces whose peculiarities stretch your intuition beyond the coordinate plane. These peculiarities include infinite dimension, function-valued vectors, and matrix-shaped vectors. Without a stock of concrete models, the abstract axioms feel like an empty shell. With them, the axioms become a genuinely useful classification device. The big picture is this: a vector space is any set equipped with an addition and a scalar multiplication. These operations satisfy eight or ten axioms. The axioms include commutativity, associativity, and distributivity. They also require a zero vector and additive inverses. Finally, they require compatibility of scalar multiplication. All this happens over a fixed field F\mathbb{F}, usually R\mathbb{R} or C\mathbb{C}. The examples below are the canonical models for every abstract theorem. These include Fn\mathbb{F}^n, polynomial spaces P(F)\mathcal{P}(\mathbb{F}) and Pn(F)\mathcal{P}_n(\mathbb{F}), function spaces F(S,F)\mathcal{F}(S,\mathbb{F}), matrix spaces Mm×n(F)M_{m\times n}(\mathbb{F}), and sequence spaces F∞\mathbb{F}^{\infty}. Every abstract theorem is secretly designed to describe them. We verify the axioms carefully for the least obvious cases. We also highlight where the verification actually requires an argument rather than a glance. By the end, you should be able to recognize a vector space "in the wild." You should be able to verify the axioms efficiently. Do this by locating the correct ambient operations. You should also see why these five families recur throughout the rest of linear algebra. This continues from Subspaces, Span and Linear Combinations onward.

Key takeaways
  • A vector space is a set with addition and scalar multiplication (over a field F\mathbb{F}) satisfying ten axioms; the operations, not the elements, define the structure.
  • Fn\mathbb{F}^n, the space of nn-tuples, is the prototype; every finite-dimensional space "looks like" some Fn\mathbb{F}^n.
  • P(F)\mathcal{P}(\mathbb{F}) (all polynomials) and Pn(F)\mathcal{P}_n(\mathbb{F}) (degree ≤n\le n) are vector spaces under ordinary polynomial addition and scalar multiplication.
  • F(S,F)\mathcal{F}(S,\mathbb{F}), all functions from a set SS to F\mathbb{F}, is a vector space under pointwise operations — this includes sequence spaces as the case S=NS = \mathbb{N}.
  • Mm×n(F)M_{m\times n}(\mathbb{F}), all m×nm\times n matrices, is a vector space under entrywise addition and scaling; it is essentially Fmn\mathbb{F}^{mn} in disguise.
  • Verifying "it's a vector space" always reduces to checking closure plus the ten axioms, most of which are inherited automatically from the field F\mathbb{F}.
  • Key idea: a vector space is a pattern, not a place — arrows, polynomials, matrices, and infinite sequences are all the same pattern wearing different clothes.

Where this lives today

Function spaces F(S,F)\mathcal{F}(S,\mathbb{F}) underlie the whole of functional analysis: Hilbert spaces like L2([0,1])L^2([0,1]) and sequence spaces like ℓ2\ell^2 are exactly vector spaces of the F(S,F)\mathcal{F}(S,\mathbb{F})/F∞\mathbb{F}^\infty type equipped with an additional inner product, used throughout quantum mechanics (state spaces of quantum systems are complex vector spaces of exactly this flavor) and signal processing (a digital signal is literally an element of a sequence space). Polynomial spaces Pn(F)\mathcal{P}_n(\mathbb{F}) are the natural home for polynomial interpolation and numerical approximation, and their finite-dimensionality is what makes methods like least-squares polynomial fitting computationally tractable. Matrix spaces Mm×n(F)M_{m\times n}(\mathbb{F}) are the working arena of numerical linear algebra, statistics (design matrices, covariance matrices), computer graphics (transformation matrices), and machine learning (weight matrices of neural network layers, which are literally added and scaled during optimization exactly as the axioms prescribe). Coordinate spaces Fn\mathbb{F}^n, finally, appear everywhere as the target of coordinates once a basis is chosen — a theme developed fully once Subspaces, Span and Linear Combinations and later bases and dimension are introduced — and F∞\mathbb{F}^{\infty}-type constructions appear in cryptography (infinite-dimensional code and lattice constructions) and in the theory of formal power series used in combinatorics and computer science.

Ask the tutor: Can you show me step-by-step how to check whether the set of matrices with trace zero is a vector space?

DefinitionVector space over a field F\mathbb{F}

Let F\mathbb{F} be a field (throughout, take F=R\mathbb{F} = \mathbb{R} or F=C\mathbb{F} = \mathbb{C} unless stated otherwise). A vector space over F\mathbb{F} is a set VV together with two operations,

+:V×V→V(vector addition),⋅:F×V→V(scalar multiplication),+ : V \times V \to V \qquad \text{(vector addition)}, \qquad \cdot : \mathbb{F} \times V \to V \qquad \text{(scalar multiplication)},

such that for all u,v,w∈Vu, v, w \in V and all a,b∈Fa, b \in \mathbb{F}: 1. (Commutativity) u+v=v+uu + v = v + u. 2. (Associativity of addition) (u+v)+w=u+(v+w)(u+v)+w = u+(v+w). 3. (Additive identity) There exists 0∈V0 \in V with v+0=vv + 0 = v for all v∈Vv \in V. 4. (Additive inverses) For each v∈Vv \in V there exists −v∈V-v \in V with v+(−v)=0v + (-v) = 0. 5. (Multiplicative identity) 1⋅v=v1 \cdot v = v for all v∈Vv \in V, where 11 is the multiplicative identity of F\mathbb{F}. 6. (Associativity of scalar multiplication) (ab)v=a(bv)(ab)v = a(bv). 7. (Distributivity over vector addition) a(u+v)=au+ava(u+v) = au + av. 8. (Distributivity over scalar addition) (a+b)v=av+bv(a+b)v = av + bv. Elements of VV are called vectors; elements of F\mathbb{F} are called scalars. A subset does not automatically inherit vector space structure — one must specify which addition and scalar multiplication are in use, since the same underlying set can support inequivalent operations.

Intuition

The single biggest conceptual jump in linear algebra is realizing that "vector" does not mean "arrow with a tail and a head." It means "any object that can be added to another object of its kind, and scaled by a number, in a way consistent with the eight rules above." Once you accept that, a polynomial 3x2−5x+13x^2 - 5x + 1 is a vector exactly as legitimately as (3,−5,1)∈R3(3,-5,1) \in \mathbb{R}^3 is — and in fact these two objects behave identically under addition and scaling, because a degree-≤2\le 2 polynomial is completely determined by its coefficient triple. This suggests a picture: think of each vector space as a set of "slots" that can hold numbers, with addition and scaling acting slot-by-slot. In Fn\mathbb{F}^n the slots are the nn coordinate positions. In Pn(F)\mathcal{P}_n(\mathbb{F}) the slots are the coefficients of 1,x,…,xn1, x, \dots, x^n. In Mm×n(F)M_{m\times n}(\mathbb{F}) the slots are the mnmn matrix entries. In a function space F(S,F)\mathcal{F}(S,\mathbb{F}) the "slots" are indexed by the (possibly infinite, possibly uncountable) set SS, one slot per point of SS — and a sequence space F∞\mathbb{F}^\infty is just the special case S=NS = \mathbb{N}. The unifying image is a coordinate rack: pick an index set, hang a scalar on each hook, and add/scale hook-by-hook.

Figure
Same pattern, different "shape" of index set: a finite rack (F3\mathbb{F}^3), a polynomial rack (P2\mathcal{P}_2), and an infinite rack (F∞\mathbb{F}^\infty)
F³: (2, −1, 4) 2 −1 4 hooks: e₁ e₂ e₃ P₂: 3 − 2x + x² 3 −2 1 hooks: 1 x x² F^∞: (1, 1/2, 1/3, …) 1 1/2 1/3 … hooks indexed by ℕ, no last one Addition and scaling always act "hook-by-hook" — same pattern, different index set. Finite racks (F³, P₂, M_(m×n)) are all secretly F^n for some n; F^∞ and F(S,F) are genuinely infinite.

The picture also flags the one genuinely new phenomenon in this article: when the index set is infinite (as in F∞\mathbb{F}^\infty or F(S,F)\mathcal{F}(S,\mathbb{F}) for infinite SS), there is no finite list of "coordinates," and the space cannot be spanned by finitely many vectors. This is the first hint of the distinction between finite- and infinite-dimensional spaces, a theme picked up in earnest in Subspaces, Span and Linear Combinations.

TheoremThe five canonical constructions are vector spaces

Let F\mathbb{F} be a field. Each of the following is a vector space over F\mathbb{F}, under the stated operations.

(a) Coordinate space Fn\mathbb{F}^n. For a fixed positive integer nn, let

Fn={(x1,…,xn):xi∈F for i=1,…,n},\mathbb{F}^n = \{(x_1,\dots,x_n) : x_i \in \mathbb{F} \text{ for } i=1,\dots,n\},

with addition (x1,…,xn)+(y1,…,yn)=(x1+y1,…,xn+yn)(x_1,\dots,x_n)+(y_1,\dots,y_n) = (x_1+y_1,\dots,x_n+y_n) and scalar multiplication a(x1,…,xn)=(ax1,…,axn)a(x_1,\dots,x_n) = (ax_1,\dots,ax_n).

(b) Polynomial spaces. Let P(F)\mathcal{P}(\mathbb{F}) be the set of all polynomials with coefficients in F\mathbb{F},

P(F)={a0+a1x+⋯+amxm:m≥0, ai∈F},\mathcal{P}(\mathbb{F}) = \Big\{ a_0 + a_1 x + \cdots + a_m x^m : m \ge 0,\ a_i \in \mathbb{F} \Big\},

with the usual polynomial addition and scalar multiplication. For fixed n≥0n \ge 0, let Pn(F)⊆P(F)\mathcal{P}_n(\mathbb{F}) \subseteq \mathcal{P}(\mathbb{F}) be the polynomials of degree at most nn (including the zero polynomial), with the same operations. Both are vector spaces over F\mathbb{F}.

(c) Function spaces. For any nonempty set SS, let

F(S,F)={f:S→F}\mathcal{F}(S,\mathbb{F}) = \{ f : S \to \mathbb{F} \}

with pointwise operations: (f+g)(s)=f(s)+g(s)(f+g)(s) = f(s)+g(s) and (af)(s)=a⋅f(s)(af)(s) = a\cdot f(s) for all s∈Ss \in S. Then F(S,F)\mathcal{F}(S,\mathbb{F}) is a vector space over F\mathbb{F}.

(d) Matrix spaces. For fixed positive integers m,nm,n, let Mm×n(F)M_{m\times n}(\mathbb{F}) be the set of all m×nm \times n matrices with entries in F\mathbb{F}, with entrywise addition (A+B)ij=Aij+Bij(A+B)_{ij} = A_{ij}+B_{ij} and entrywise scaling (aA)ij=aAij(aA)_{ij} = aA_{ij}. Then Mm×n(F)M_{m\times n}(\mathbb{F}) is a vector space over F\mathbb{F}.

(e) Sequence spaces. Let F∞={(x1,x2,x3,… ):xi∈F for all i∈N}\mathbb{F}^{\infty} = \{ (x_1,x_2,x_3,\dots) : x_i \in \mathbb{F} \text{ for all } i \in \mathbb{N}\}, with termwise addition (xi)+(yi)=(xi+yi)(x_i)+(y_i) = (x_i+y_i) and scaling a(xi)=(axi)a(x_i) = (ax_i). Then F∞\mathbb{F}^\infty is a vector space over F\mathbb{F}; it coincides with F(N,F)\mathcal{F}(\mathbb{N},\mathbb{F}).

Proof

Technique: for each construction, we first check closure (that ++ and scalar multiplication really land back in the set), and then verify the eight axioms; almost all of them reduce directly to the corresponding field axioms of F\mathbb{F}, applied coordinatewise, pointwise, or termwise. We do this once carefully for (c), the most general construction, and note that (a), (d), and (e) are special or structurally identical cases; then we verify (b) separately since polynomials are not literally tuples and closure of degree bounds needs a one-line argument.

Part (c): F(S,F)\mathcal{F}(S,\mathbb{F}) is a vector space. Closure. If f,g:S→Ff, g : S \to \mathbb{F} are functions and a∈Fa \in \mathbb{F}, then f+gf+g (defined by (f+g)(s)=f(s)+g(s)(f+g)(s) = f(s)+g(s)) and afaf (defined by (af)(s)=af(s)(af)(s) = a f(s)) are again functions from SS to F\mathbb{F}, since for each s∈Ss \in S the values f(s)+g(s)f(s)+g(s) and af(s)a f(s) are well-defined elements of F\mathbb{F} (using that F\mathbb{F} is closed under its own addition and multiplication). So ++ and scalar multiplication indeed map F(S,F)×F(S,F)→F(S,F)\mathcal{F}(S,\mathbb{F}) \times \mathcal{F}(S,\mathbb{F}) \to \mathcal{F}(S,\mathbb{F}) and F×F(S,F)→F(S,F)\mathbb{F} \times \mathcal{F}(S,\mathbb{F}) \to \mathcal{F}(S,\mathbb{F}) as required. Axiom 1 (commutativity). For f,g∈F(S,F)f, g \in \mathcal{F}(S,\mathbb{F}) and any s∈Ss \in S,

(f+g)(s)=f(s)+g(s)=g(s)+f(s)=(g+f)(s),(f+g)(s) = f(s) + g(s) = g(s) + f(s) = (g+f)(s),

where the middle equality is commutativity of addition in the field F\mathbb{F}. Since (f+g)(s)=(g+f)(s)(f+g)(s) = (g+f)(s) for every s∈Ss \in S, the functions f+gf+g and g+fg+f are equal as functions (two functions with the same domain are equal iff they agree at every point). Hence f+g=g+ff+g = g+f. Axiom 2 (associativity of addition). For any s∈Ss \in S,

((f+g)+h)(s)=(f+g)(s)+h(s)=(f(s)+g(s))+h(s)=f(s)+(g(s)+h(s))=(f+(g+h))(s),\big((f+g)+h\big)(s) = (f+g)(s) + h(s) = \big(f(s)+g(s)\big) + h(s) = f(s) + \big(g(s)+h(s)\big) = \big(f+(g+h)\big)(s),

using associativity in F\mathbb{F} at the middle step. Agreement at every ss gives (f+g)+h=f+(g+h)(f+g)+h = f+(g+h). Axiom 3 (additive identity). Let 0:S→F\mathbf{0} : S \to \mathbb{F} be the constant function 0(s)=0F\mathbf{0}(s) = 0_{\mathbb{F}} for all ss. Then for any ff and any ss, (f+0)(s)=f(s)+0F=f(s)(f+\mathbf{0})(s) = f(s) + 0_{\mathbb{F}} = f(s), so f+0=ff + \mathbf{0} = f. Thus 0∈F(S,F)\mathbf{0} \in \mathcal{F}(S,\mathbb{F}) is an additive identity. Axiom 4 (additive inverses). Given ff, define −f:S→F-f : S \to \mathbb{F} by (−f)(s)=−f(s)(-f)(s) = -f(s) (the field-theoretic negative). Then (f+(−f))(s)=f(s)+(−f(s))=0F=0(s)(f+(-f))(s) = f(s) + (-f(s)) = 0_{\mathbb{F}} = \mathbf{0}(s) for every ss, so f+(−f)=0f + (-f) = \mathbf{0}. Axiom 5 (multiplicative identity). For 1∈F1 \in \mathbb{F} and any ss, (1⋅f)(s)=1⋅f(s)=f(s)(1 \cdot f)(s) = 1 \cdot f(s) = f(s) by the multiplicative identity axiom in F\mathbb{F}; hence 1⋅f=f1 \cdot f = f. Axiom 6 (associativity of scalars). For a,b∈Fa,b \in \mathbb{F} and any ss,

((ab)f)(s)=(ab)f(s)=a(bf(s))=(a(bf))(s),\big((ab)f\big)(s) = (ab)f(s) = a\big(b f(s)\big) = \big(a(bf)\big)(s),

using associativity of multiplication in F\mathbb{F}. Hence (ab)f=a(bf)(ab)f = a(bf). Axiom 7 (distributivity over vector addition). For any ss,

(a(f+g))(s)=a(f(s)+g(s))=af(s)+ag(s)=(af+ag)(s),\big(a(f+g)\big)(s) = a\big(f(s)+g(s)\big) = a f(s) + a g(s) = (af+ag)(s),

using the distributive law in F\mathbb{F}. Hence a(f+g)=af+aga(f+g) = af+ag. Axiom 8 (distributivity over scalar addition). For any ss,

((a+b)f)(s)=(a+b)f(s)=af(s)+bf(s)=(af+bf)(s),\big((a+b)f\big)(s) = (a+b) f(s) = a f(s) + b f(s) = (af + bf)(s),

again using distributivity in F\mathbb{F}. Hence (a+b)f=af+bf(a+b)f = af+bf. All eight axioms hold, so F(S,F)\mathcal{F}(S,\mathbb{F}) is a vector space over F\mathbb{F}.

Part (a): Fn\mathbb{F}^n. Take S={1,2,…,n}S = \{1,2,\dots,n\} in Part (c). A function f:{1,…,n}→Ff : \{1,\dots,n\} \to \mathbb{F} is exactly the same data as a tuple (f(1),…,f(n))∈Fn(f(1),\dots,f(n)) \in \mathbb{F}^n, and pointwise addition/scaling of such functions corresponds exactly to the coordinatewise operations defining Fn\mathbb{F}^n. Since F({1,…,n},F)\mathcal{F}(\{1,\dots,n\},\mathbb{F}) is a vector space by Part (c), so is Fn\mathbb{F}^n under the stated operations (the correspondence is a relabeling, not a change of structure).

Part (e): F∞\mathbb{F}^\infty. Take S=N={1,2,3,… }S = \mathbb{N} = \{1,2,3,\dots\} in Part (c). A sequence (xi)i=1∞(x_i)_{i=1}^\infty is precisely a function x:N→Fx : \mathbb{N} \to \mathbb{F}, x(i)=xix(i) = x_i, and termwise addition/scaling of sequences is pointwise addition/scaling of the corresponding functions. Hence F∞=F(N,F)\mathbb{F}^\infty = \mathcal{F}(\mathbb{N},\mathbb{F}) literally, and it is a vector space by Part (c).

Part (d): Mm×n(F)M_{m\times n}(\mathbb{F}). Take S={1,…,m}×{1,…,n}S = \{1,\dots,m\} \times \{1,\dots,n\} in Part (c). A matrix A∈Mm×n(F)A \in M_{m\times n}(\mathbb{F}) is precisely a function A:S→FA : S \to \mathbb{F} sending (i,j)↦Aij(i,j) \mapsto A_{ij}, and entrywise addition/scaling of matrices is exactly pointwise addition/scaling of the corresponding functions on SS. Since SS is a finite set of size mnmn, this again shows Mm×n(F)M_{m\times n}(\mathbb{F}) is (after relabeling entries as a single list) essentially Fmn\mathbb{F}^{mn}, and it is a vector space by Part (c).

Part (b): P(F)\mathcal{P}(\mathbb{F}) and Pn(F)\mathcal{P}_n(\mathbb{F}). Closure for P(F)\mathcal{P}(\mathbb{F}). If p(x)=∑i=0maixip(x) = \sum_{i=0}^{m} a_i x^i and q(x)=∑i=0kbixiq(x) = \sum_{i=0}^{k} b_i x^i are polynomials (extend the shorter list of coefficients by zeros so both run to N=max⁡(m,k)N = \max(m,k)), then

p(x)+q(x)=∑i=0N(ai+bi)xi,c p(x)=∑i=0m(cai)xip(x) + q(x) = \sum_{i=0}^{N} (a_i+b_i)x^i, \qquad c\, p(x) = \sum_{i=0}^m (c a_i) x^i

are again finite sums of powers of xx with coefficients in F\mathbb{F}, i.e. again elements of P(F)\mathcal{P}(\mathbb{F}). So P(F)\mathcal{P}(\mathbb{F}) is closed under both operations. Closure for Pn(F)\mathcal{P}_n(\mathbb{F}). If deg⁡p≤n\deg p \le n and deg⁡q≤n\deg q \le n (allowing deg⁡(0)=−∞\deg(0) = -\infty by convention, so the zero polynomial is included), write p(x)=∑i=0naixip(x)=\sum_{i=0}^n a_ix^i, q(x)=∑i=0nbixiq(x)=\sum_{i=0}^n b_i x^i (padding with zero coefficients as needed up to index nn). Then p+q=∑i=0n(ai+bi)xip+q = \sum_{i=0}^n (a_i+b_i)x^i has degree at most nn, since no coefficient with index exceeding nn appears; and cp=∑i=0n(cai)xicp = \sum_{i=0}^n (ca_i)x^i likewise has degree at most nn. So Pn(F)\mathcal{P}_n(\mathbb{F}) is closed under both operations. The eight axioms. Identify a polynomial p(x)=∑i=0naixip(x) = \sum_{i=0}^n a_i x^i of degree ≤n\le n with its coefficient tuple (a0,a1,…,an)∈Fn+1(a_0,a_1,\dots,a_n) \in \mathbb{F}^{n+1}. Under this identification, polynomial addition and scalar multiplication correspond exactly to coordinatewise addition and scaling in Fn+1\mathbb{F}^{n+1} (this is immediate from the definitions above: the coefficient of xix^i in p+qp+q is ai+bia_i+b_i, and in cpcp is caica_i). Since Fn+1\mathbb{F}^{n+1} is a vector space by Part (a), and the identification carries the operations exactly, Pn(F)\mathcal{P}_n(\mathbb{F}) satisfies all eight axioms, with additive identity the zero polynomial and additive inverse of pp given by −p=∑(−ai)xi-p = \sum (-a_i)x^i. For P(F)\mathcal{P}(\mathbb{F}) itself, there is no single finite nn bounding all degrees, so this exact tuple identification does not apply globally; instead verify the axioms directly, coefficient-by-coefficient, exactly as in the proof of Part (c) but with the index set being the (varying, finite for each polynomial) set of coefficient positions. Concretely: given any finitely many polynomials appearing in an axiom (at most three, e.g. p,q,rp,q,r in associativity), let NN be the maximum of their degrees and pad all coefficient lists with zeros to length N+1N+1; the axiom then reduces to the corresponding axiom in FN+1\mathbb{F}^{N+1}, already established. Since NN is finite for any given instance of the axiom, this suffices to verify the axiom for all polynomials. The zero polynomial (deg⁡=−∞\deg = -\infty, all coefficients 00) is the additive identity, and −p-p (negate every coefficient) is the additive inverse of pp. This completes the verification of all five parts. ■\blacksquare

∎

Worked examples

Example 1 (a non-obvious subset check inside P3(R)\mathcal{P}_3(\mathbb{R}), contrasted with M2×2M_{2\times 2}). Consider V=P3(R)V = \mathcal{P}_3(\mathbb{R}), polynomials of degree at most 33, and the two specific vectors

p(x)=2x3−5x+7,q(x)=−x3+4x2+x−1.p(x) = 2x^3 - 5x + 7, \qquad q(x) = -x^3 + 4x^2 + x - 1.

Let's compute p+3qp + 3q directly from the axioms/operations, and simultaneously verify closure is doing real work (not just formal bookkeeping).

3q(x)=−3x3+12x2+3x−3.3q(x) = -3x^3 + 12x^2 + 3x - 3.
p(x)+3q(x)=(2x3−5x+7)+(−3x3+12x2+3x−3)=(2−3)x3+12x2+(−5+3)x+(7−3).p(x) + 3q(x) = (2x^3 - 5x + 7) + (-3x^3+12x^2+3x-3) = (2-3)x^3 + 12x^2 + (-5+3)x + (7-3).
p(x)+3q(x)=−x3+12x2−2x+4.p(x) + 3q(x) = -x^3 + 12x^2 -2x + 4.

This has degree exactly 3≤33 \le 3, confirming it lies in P3(R)\mathcal{P}_3(\mathbb{R}) — closure holds even though the leading coefficients 22 and −3-3 did not cancel (had we chosen coefficients that canceled the x3x^3 term, we'd land in P2(R)⊂P3(R)\mathcal{P}_2(\mathbb{R}) \subset \mathcal{P}_3(\mathbb{R}), still fine). The key point demonstrated: closure of Pn\mathcal{P}_n under addition/scaling is not automatic bookkeeping — it genuinely relies on the fact that adding polynomials of degree ≤n\le n cannot produce a term of degree >n> n, since no such terms exist to add in the first place. Under the tuple correspondence of the proof, p↔(7,−5,0,2)p \leftrightarrow (7,-5,0,2), q↔(−1,1,4,−1)q \leftrightarrow (-1,1,4,-1), and

p+3q↔(7,−5,0,2)+3(−1,1,4,−1)=(7,−5,0,2)+(−3,3,12,−3)=(4,−2,12,−1),p + 3q \leftrightarrow (7,-5,0,2) + 3(-1,1,4,-1) = (7,-5,0,2) + (-3,3,12,-3) = (4,-2,12,-1),

matching the coefficient list (4,−2,12,−1)(4,-2,12,-1) of −x3+12x2−2x+4-x^3+12x^2-2x+4 read low-to-high degree. This confirms the identification with R4\mathbb{R}^4 used in the theorem's proof is not just conceptual but numerically exact. Now contrast with M2×2(R)M_{2\times 2}(\mathbb{R}): take

A=(2−105),B=(34−21).A = \begin{pmatrix} 2 & -1 \\ 0 & 5 \end{pmatrix}, \qquad B = \begin{pmatrix} 3 & 4 \\ -2 & 1 \end{pmatrix}.

Then

2A−B=(4−2010)−(34−21)=(1−629),2A - B = \begin{pmatrix} 4 & -2 \\ 0 & 10 \end{pmatrix} - \begin{pmatrix} 3 & 4 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & -6 \\ 2 & 9 \end{pmatrix},

computed entrywise exactly as the theorem's Part (d) prescribes: (2A−B)ij=2Aij−Bij(2A-B)_{ij} = 2A_{ij} - B_{ij} for each of the four positions (i,j)(i,j). Both computations — one in a polynomial space, one in a matrix space — are the same linear combination operation, differing only in how the "coordinates" are laid out (as coefficients of powers of xx, versus as entries of a rectangular array).

Example 2 (an infinite-dimensional computation in F∞\mathbb{F}^\infty, showing what does and does not make sense). Let F=R\mathbb{F} = \mathbb{R}, and consider two sequences in R∞\mathbb{R}^\infty:

u=(1,12,13,14,… )=(1n)n≥1,v=(1,−1,1,−1,… )=((−1)n+1)n≥1.u = (1, \tfrac12, \tfrac13, \tfrac14, \dots) = \left(\tfrac1n\right)_{n\ge 1}, \qquad v = (1,-1,1,-1,\dots) = \big((-1)^{n+1}\big)_{n \ge 1}.

By definition of the vector space operations on F∞\mathbb{F}^\infty (Part (e) of the theorem), for u+2vu + 2v we compute termwise:

(u+2v)n=1n+2(−1)n+1.(u+2v)_n = \frac1n + 2(-1)^{n+1}.

So the first four terms are

n=1: 1+2(1)=3,n=2: 12+2(−1)=−32,n=3: 13+2(1)=73,n=4: 14+2(−1)=−74,n=1: \ 1 + 2(1) = 3, \qquad n=2: \ \tfrac12 + 2(-1) = -\tfrac32, \qquad n=3: \ \tfrac13 + 2(1) = \tfrac73, \qquad n=4: \ \tfrac14 + 2(-1) = -\tfrac74,

giving u+2v=(3,−32,73,−74,… )u + 2v = \left(3, -\tfrac32, \tfrac73, -\tfrac74, \dots\right), an entirely well-defined element of R∞\mathbb{R}^\infty: the vector space axioms guarantee this termwise recipe produces a genuine sequence, with no convergence or boundedness required, since R∞\mathbb{R}^\infty places no analytic condition on the sequences — only pointwise (termwise) algebra matters. Notice what would go wrong if we tried to ask "what is u⋅vu \cdot v" (entrywise product) or "what is lim⁡nun\lim_n u_n" as if it were part of the vector space structure: neither operation is part of the vector space axioms. The vector space structure on F∞\mathbb{F}^\infty knows only about ++ and scalar multiplication; it says nothing about products of two vectors or about limits. This is a common point of confusion: u+2vu + 2v is a legitimate vector space computation, but statements like "uu converges" or "uvuv is bounded" are extra structure not supplied by the vector space axioms alone (they belong to later, richer objects such as sequence spaces with norms, e.g. ℓ2\ell^2, which are vector spaces plus additional structure, not just vector spaces).

Counterexamplewhy closure must actually be checked — dropping it breaks everything

The theorem's proof of closure was not cosmetic. Here is what happens if we alter one of the five constructions just enough to destroy closure, while leaving every other axiom looking fine on the surface. Let W={p∈P(R):deg⁡p=3}W = \{ p \in \mathcal{P}(\mathbb{R}) : \deg p = 3 \}, the set of polynomials of degree exactly 33 (not "at most 33"), with the same polynomial addition and scalar multiplication as in P(R)\mathcal{P}(\mathbb{R}). Is WW a vector space under these operations? Take p(x)=x3+xp(x) = x^3 + x and q(x)=−x3+2xq(x) = -x^3 + 2x, both in WW (each has degree exactly 33, since the leading coefficient ±1≠0\pm1 \ne 0). Then

p(x)+q(x)=(x3+x)+(−x3+2x)=3x,p(x) + q(x) = (x^3+x) + (-x^3+2x) = 3x,

which has degree 11, not 33. So p+q∉Wp+q \notin W: the set WW is not closed under the very addition operation we were trying to use. Since ++ does not even map W×W→WW \times W \to W, the pair (W,+,⋅)(W,+,\cdot) fails to be a vector space before we can even ask about axioms 1–8 — the addition operation is not well-defined as a map into WW. (Also, WW contains no additive identity: the zero polynomial has degree −∞-\infty, not 33, so 0∉W0 \notin W, and axiom 3 fails outright even considered in isolation.) This shows precisely why the theorem's proof insisted on checking closure before the axioms: in Pn(F)\mathcal{P}_n(\mathbb{F}) (degree at most nn), the degree bound is preserved by addition because "at most nn" is closed under cancellation-or-no-cancellation of leading terms, whereas "exactly nn" is not — a leading term can cancel, silently dropping the degree. Requiring "at most nn" rather than "exactly nn" is the hypothesis that makes Pn(F)\mathcal{P}_n(\mathbb{F}) a genuine vector space, and dropping it (replacing "at most" by "exactly") breaks the conclusion.

Scope, edge cases & common traps

SituationWhat to checkCommon mistake
Pn(F)\mathcal{P}_n(\mathbb{F}) vs. degree-exactly-nn polynomialsClosure under addition (leading terms can cancel)Assuming "degree nn" sets are automatically closed
Fn\mathbb{F}^n vs. F∞\mathbb{F}^\inftyWhether the index set is finite or infiniteTreating F∞\mathbb{F}^\infty as if it had "coordinates n+1,…n+1,\dots" that eventually stop
Function space F(S,F)\mathcal{F}(S,\mathbb{F}) with SS infinite/uncountableOnly pointwise algebra is guaranteed; no topology, no continuity, no boundednessAssuming F(S,R)\mathcal{F}(S,\mathbb{R}) vectors are automatically continuous, bounded, or integrable
Matrix spaces Mm×n(F)M_{m\times n}(\mathbb{F})Addition only defined for same-shape matricesTrying to add an m×nm\times n and a p×qp \times q matrix with (m,n)≠(p,q)(m,n)\ne(p,q)
Real vs. complex scalarsThe same set can be a vector space over R\mathbb{R} or over C\mathbb{C}, with different behaviorForgetting to specify F\mathbb{F}, e.g. treating C\mathbb{C} as "the same" 22-dimensional space over R\mathbb{R} and over C\mathbb{C} (dimension differs: 22 vs. 11)
Sequence space F∞\mathbb{F}^\infty vs. ℓ2,c0\ell^2, c_0, etc.F∞\mathbb{F}^\infty has no convergence/boundedness restrictionConfusing F∞\mathbb{F}^\infty (all sequences) with subspaces defined by analytic conditions

The mathematician behind it

Origins · 1 minGiuseppe Peano and how this was discovered
GP
Giuseppe Peano
1858–1932 · axiomatic foundations of mathematics, Turin

Giuseppe Peano, an Italian mathematician best known to most students for the Peano axioms of arithmetic, was also the first person to write down an essentially modern, fully explicit list of vector space axioms, in his 1888 book Calcolo Geometrico. Building on Hermann Grassmann's earlier (but less formally axiomatic) theory of extension, Peano isolated exactly the closure and algebraic properties needed for "linear combination" to make sense abstractly, independent of whether the underlying objects were geometric vectors, forces, or — as this article emphasizes — polynomials, functions, or matrices. His formulation anticipated by decades the general treatment that became standard only after further work by Hermann Weyl (1918) and the Bourbaki school in the 20th century, by which point the axioms had stabilized into the form given in the definition above.

History
- 1844: Hermann Grassmann publishes Die lineale Ausdehnungslehre, introducing an early, largely unread theory of nn-dimensional linear extension.
- 1888: Giuseppe Peano publishes Calcolo Geometrico, giving the first fully explicit axiomatic definition of a real vector space.
- 1918: Hermann Weyl's Raum, Zeit, Materie popularizes an axiomatic vector space treatment in the context of relativity, reaching a wider audience of physicists and mathematicians.
- 1920s–1930s: Stefan Banach and others extend the vector space idea to infinite-dimensional function spaces equipped with norms, directly building on function-space examples like those in this article.
- 1930s–1940s: The Bourbaki collective standardizes the modern axiom list and terminology used across contemporary linear algebra texts.
Is every vector space literally equal to one of these five examples?

Not literally, but every finite-dimensional vector space over F\mathbb{F} is isomorphic to some Fn\mathbb{F}^n — meaning there is a structure-preserving bijection between them, a fact proved later once bases and dimension are developed. Infinite-dimensional spaces (like P(F)\mathcal{P}(\mathbb{F}), sequence spaces, or general function spaces) are not isomorphic to any Fn\mathbb{F}^n, but the "index set" picture from this article's intuition section still applies with an infinite index set. So these five families are not an exhaustive list of all vector spaces, but they are a representative list: essentially every vector space you meet in an introductory course is either literally one of these, a subspace of one of these, or isomorphic to one of these.

Why does Pn(F)\mathcal{P}_n(\mathbb{F}) include the zero polynomial, and why is that necessary?

Every vector space must contain an additive identity element (axiom 3), and in Pn(F)\mathcal{P}_n(\mathbb{F}) that identity is the zero polynomial 00. If the zero polynomial were excluded, axiom 3 would fail outright, since there would be no vector z∈Pn(F)z \in \mathcal{P}_n(\mathbb{F}) with p+z=pp + z = p for all pp. Conventionally, the zero polynomial is assigned degree −∞-\infty (or is simply declared to have "no degree" but to satisfy deg⁡(0)≤n\deg(0) \le n for every nn by convention), precisely so that it can be included in Pn(F)\mathcal{P}_n(\mathbb{F}) for every n≥0n \ge 0 without contradiction.

What exactly goes wrong if I try to add two matrices of different shapes?

The addition operation on Mm×n(F)M_{m\times n}(\mathbb{F}) is defined entrywise, (A+B)ij=Aij+Bij(A+B)_{ij} = A_{ij}+B_{ij}, which requires both AA and BB to have entries indexed by the same pairs (i,j)(i,j) with 1≤i≤m1 \le i \le m, 1≤j≤n1 \le j \le n. If AA is 2×32\times 3 and BB is 3×23\times 2, there is no sensible entrywise pairing — AA has an entry A13A_{13} but BB has no entry B13B_{13} (since BB only has 22 columns). So A+BA+B is simply undefined, not merely "wrong": the operation ++ on Mm×n(F)M_{m\times n}(\mathbb{F}) only ever takes two same-shape matrices as input, by the very definition of the set Mm×n(F)M_{m\times n}(\mathbb{F}) as matrices of one fixed shape.

Is F∞\mathbb{F}^\infty the same as RN\mathbb{R}^\mathbb{N} or the space of convergent sequences?

F∞\mathbb{F}^\infty as defined here (all sequences (x1,x2,… )(x_1,x_2,\dots) with xi∈Fx_i \in \mathbb{F}) is exactly the same set as F(N,F)\mathcal{F}(\mathbb{N},\mathbb{F}), sometimes written FN\mathbb{F}^{\mathbb{N}}, with no convergence condition imposed. It is not the same as the space of convergent sequences, or the space ℓ2\ell^2 of square-summable sequences, or c0c_0 (sequences converging to 00) — those are all proper subsets of F∞\mathbb{F}^\infty that happen to also be vector spaces in their own right (in fact they are subspaces of F∞\mathbb{F}^\infty), because convergence and summability conditions are each preserved under addition and scalar multiplication. But F∞\mathbb{F}^\infty itself imposes no such restriction.

Does the choice of field F\mathbb{F} change whether something is a vector space?

Yes, fundamentally. The same underlying set can be a vector space over different fields with different properties. For instance, C\mathbb{C} is a vector space over C\mathbb{C} itself (of dimension 11, foreshadowing later material) but is also a vector space over R\mathbb{R} (of dimension 22, with basis {1,i}\{1,i\}), using the same addition but restricting scalars to real numbers. All five constructions in this article — Fn\mathbb{F}^n, P(F)\mathcal{P}(\mathbb{F}), F(S,F)\mathcal{F}(S,\mathbb{F}), Mm×n(F)M_{m\times n}(\mathbb{F}), F∞\mathbb{F}^\infty — genuinely depend on which field F\mathbb{F} is fixed, and the proofs given work verbatim for any field, not just R\mathbb{R} or C\mathbb{C}.

Why bother proving these five are vector spaces instead of just asserting it?

Because the vector space axioms are not automatically satisfied by every set with an addition-like operation — as the counterexample with degree-exactly-33 polynomials shows, seemingly natural constructions can fail closure or another axiom. Proving the five canonical examples rigorously accomplishes two things: it demonstrates the axiom-checking technique (reduce pointwise/entrywise/coefficientwise to the corresponding field axiom) that you will reuse whenever you meet a new candidate vector space, and it certifies that every future theorem phrased for "a vector space VV" really does apply to polynomials, matrices, and sequences, not merely to geometric arrows.

How does this connect to what comes after — subspaces and span?

Once you know Fn\mathbb{F}^n, Pn(F)\mathcal{P}_n(\mathbb{F}), F(S,F)\mathcal{F}(S,\mathbb{F}), Mm×n(F)M_{m\times n}(\mathbb{F}), and F∞\mathbb{F}^\infty are vector spaces, the natural next question is which subsets of these spaces are themselves vector spaces under the inherited operations — this is the notion of a subspace. For example, Pn(F)\mathcal{P}_n(\mathbb{F}) is itself a subspace of P(F)\mathcal{P}(\mathbb{F}), and the symmetric matrices form a subspace of Mn×n(F)M_{n\times n}(\mathbb{F}). The related notion of span — the set of all linear combinations of a given list of vectors — is what lets us build up subspaces from finite lists of vectors inside any of these five examples, and is the essential next tool for measuring the "size" (dimension) of a vector space.

Before this

Where this leads