Vector Space Axioms: The Ten Rules

The Idea of Abstraction

\mathbb{R}^n

can be added entry by entry and scaled by real numbers. Polynomials can be added and scaled. Matrices can be added and scaled. Continuous functions on an interval can be added and scaled. In each case, the same algebraic patterns appear: addition is commutative and associative, scaling distributes over sums, a zero element absorbs addition, and scaling by

1

leaves every object unchanged.

A vector space is the formal extraction of these patterns. Rather than proving results separately for columns, polynomials, matrices, and functions, the axioms identify the common thread. Anything proved from the axioms alone — and that includes the entire theory of linear independence, span, basis, and dimension — holds in every setting where the axioms are satisfied.

The Ten Axioms

A vector space over a field

\mathbb{F}

is a set

V

together with two operations — vector addition (

\mathbf{u} + \mathbf{v}

) and scalar multiplication (

c\mathbf{v}

) — satisfying the following ten axioms. For all

\mathbf{u}, \mathbf{v}, \mathbf{w} \in V

and all scalars

c, d \in \mathbb{F}

:

Vector Space Axioms

\begin{aligned} &\text{For all } \mathbf{u}, \mathbf{v}, \mathbf{w} \in V \text{ and all } c, d \in \mathbb{F}: \\ &(1)\ \mathbf{u} + \mathbf{v} \in V \quad (2)\ \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \\ &(3)\ (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \\ &(4)\ \exists\, \mathbf{0} \in V: \mathbf{v} + \mathbf{0} = \mathbf{v} \\ &(5)\ \exists\, -\mathbf{v} \in V: \mathbf{v} + (-\mathbf{v}) = \mathbf{0} \\ &(6)\ c\mathbf{v} \in V \quad (7)\ c(d\mathbf{v}) = (cd)\mathbf{v} \\ &(8)\ c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} \\ &(9)\ (c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v} \quad (10)\ 1\mathbf{v} = \mathbf{v} \end{aligned}

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Addition Axioms

Closure under addition:

\mathbf{u} + \mathbf{v} \in V

.

Commutativity:

\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}

.

Associativity:

(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})

.

Zero vector: there exists an element

\mathbf{0} \in V

such that

\mathbf{v} + \mathbf{0} = \mathbf{v}

for every

\mathbf{v} \in V

.

Additive inverse: for every

\mathbf{v} \in V

, there exists

-\mathbf{v} \in V

such that

\mathbf{v} + (-\mathbf{v}) = \mathbf{0}

.

Scalar Multiplication Axioms

Closure under scalar multiplication:

c\mathbf{v} \in V

.

Associativity of scalars:

c(d\mathbf{v}) = (cd)\mathbf{v}

.

Distributivity over vector addition:

c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}

.

Distributivity over scalar addition:

(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}

.

Multiplicative identity:

1\mathbf{v} = \mathbf{v}

.

A set satisfying all ten is a vector space. A set violating even one is not.

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Vector spaces · definition

The ten axioms

Five govern addition, four govern scalar multiplication, and one ties the scalar field to the vectors. Nothing here mentions arrows, coordinates or dimension — that is the point of the list.

10axioms

Addition5

1

ClosureNon-examples

\mathbf{u} + \mathbf{v} \in V

for all

\mathbf{u}, \mathbf{v} \in V

Addition may not leave the set.

Addition may not leave the set. The most frequently violated axiom in practice — a set can look like a vector space and fail here alone, which is what disqualifies most of the non-examples.

2

Commutativity

\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}

for all

\mathbf{u}, \mathbf{v} \in V

Order does not matter.

Order does not matter. Worth noticing this is assumed rather than derived — matrix multiplication shows what an operation without it looks like.

3

Associativity

(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})

for all

\mathbf{u}, \mathbf{v}, \mathbf{w} \in V

Grouping does not matter, so a sum of many vectors needs no parentheses…

Grouping does not matter, so a sum of many vectors needs no parentheses at all. This is what makes linear combinations well defined as written.

4

Zero vectorConsequences

\exists\, \mathbf{0} \in V : \mathbf{v} + \mathbf{0} = \mathbf{v}

one such element, for every

\mathbf{v}

An additive identity.

An additive identity. The axiom asserts existence only — uniqueness is a consequence, proved in two lines by assuming two zeros and adding them to each other.

5

Additive inverseConsequences

\forall\, \mathbf{v} \;\exists\, -\mathbf{v} : \mathbf{v} + (-\mathbf{v}) = \mathbf{0}

one inverse per element

Every vector can be undone.

Every vector can be undone. Together with the four above this makes $V$ an abelian group under addition — the scalar axioms are what turn a group into a vector space.

Scalar multiplication4

6

ClosureNon-examples

c\mathbf{v} \in V

for all scalars

c

and

\mathbf{v} \in V

Scaling may not leave the set either.

Scaling may not leave the set either. Checking this and closure under addition is the whole test for whether a subset is a subspace — the other eight are inherited.

7

Associativity of scalars

c(d\mathbf{v}) = (cd)\mathbf{v}

for all scalars

c, d

Scaling twice equals scaling once by the product.

Scaling twice equals scaling once by the product. The two multiplications are different operations — $cd$ happens in the field, $c(d\mathbf{v})$ happens in $V$ — and the axiom is the claim that they agree.

8

Distributivity over vectors

c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}

for all scalars

c

Scaling spreads over a sum of vectors.

Scaling spreads over a sum of vectors. This and the next axiom are exactly what linearity demands of a transformation — the definition is these two conditions and nothing more.

9

Distributivity over scalars

(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}

for all scalars

c, d

The mirror of the axiom above, spreading over a sum of scalars instead.

The mirror of the axiom above, spreading over a sum of scalars instead. Both are needed and neither implies the other.

Tying the two together1

10

Multiplicative identityScalar field

1\mathbf{v} = \mathbf{v}

1

the identity of the scalar field

The one that looks redundant and is not.

The one that looks redundant and is not. Define scaling so that every $c\mathbf{v}$ equals $\mathbf{0}$ and the other nine axioms all hold — this axiom is the only thing ruling that out, and it is what ties $V$ to its field of scalars.

A set satisfying all ten is a vector space whatever its elements are — polynomials, matrices, functions, sequences. Everything proved from these applies to all of them at once, which is the whole return on stating them abstractly.

The ten axioms·/linear-algebra/vector-spaces/axiomsLearn Math Class

Vector Space Axiom Notation

Notation

Vector Space Axiom Notation

The axioms are written almost entirely in reused symbols. A digit stands for two objects, juxtaposition stands for two multiplications, and one plus sign is being defined by another.

The script letters

\mathcal{P}_n

and

\mathcal{B}

and their calligraphic convention are set out at basis; the braces in

\{\mathbf{0}\}

and what separates a set from its element at subspaces;

\dim

with its field subscript, where the choice of

\mathbb{F}

changes the answer, at dimension.

0\mathbf{v} = \mathbf{0}, \qquad c\mathbf{0} = \mathbf{0}

the scalar zero times v is the zero vector; c times the zero vector is the zero vector

Two different objects are written with the same digit, and only the weight of the type separates them. The upright

0

is a number in the field; the bold

\mathbf{0}

is an element of

V

, which might be a tuple, a matrix or the function that is zero everywhere. The line is worth stating precisely because the two are not the same thing.

CasesNeither statement is an axiom. Both are proved from the ten, and the proof of the first turns on reading

(0 + 0)

as addition in the field and

0\mathbf{v} + 0\mathbf{v}

as addition in the space — the same argument written without the weight distinction says nothing at all.

Also writtenSome texts use

\vec{0}

or an underlined zero, and handwriting has no bold at all, which is where the distinction is most often lost.

Do not confuseReading the equations as trivial. They assert that a scalar and a vector, both called zero, interact in the expected way, which is a claim about the two operations rather than about arithmetic.

Same glyph elsewhereThe zero vector inside braces,

\{\mathbf{0}\}

, is a set rather than an element, and the difference is set out at subspaces.

c(d\mathbf{v}) = (cd)\mathbf{v}

c times the quantity d times v equals the quantity c d, times v

Neither multiplication is written with a symbol, and they are not the same operation. Inside the right-hand parentheses,

cd

is multiplication of two scalars within the field. Everywhere else, juxtaposition means scaling a vector — an operation that takes one scalar and one vector and returns a vector. The axiom claims the two agree, and it has to be assumed because nothing forces it.

CasesThe same doubling appears in

(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}

, where the plus on the left is addition in

\mathbb{F}

and the plus on the right is addition in

V

. Both distributive axioms are statements that a field operation and a space operation line up.

Also writtenWritten with an explicit dot,

c \cdot (d \cdot \mathbf{v})

, in treatments that want the two multiplications visibly separated. The dot does not distinguish them either.

Do not confuseTreating the parentheses as removable, as they would be for ordinary numbers. Here they record which operation runs first, and the axiom exists precisely because dropping them assumes the answer.

Same glyph elsewhereThe same reuse of one glyph for operations in two different spaces governs the definition of a linear map at transformation properties.

(f + g)(x) = f(x) + g(x)

the function f plus g, evaluated at x, equals f of x plus g of x

The two plus signs are not doing the same work. The one on the left is being defined; there is no prior meaning to adding two functions. The one on the right is ordinary addition of real numbers, already available. The line is a definition disguised as an identity, and it reads as an identity only if the difference is missed.

CasesThe parentheses on the left group

f + g

into a single function, which is then applied to

x

. Reading them as multiplication or as

f + g(x)

changes the statement into something false.

Also writtenCalled pointwise addition, the word recording that the definition is made one input at a time and inherits its properties from the numbers.

Do not confuseAssuming a definition of this shape always produces a vector space. It supplies the operations; closure still has to be checked separately, which is what makes continuity a genuine hypothesis in

C[a, b]

.

(u_1, u_2) \oplus (v_1, v_2) = (u_1 + v_1, 0)

u circle-plus v is defined as the pair u one plus v one, zero

The circle is a warning. It marks an operation that has been stipulated for the occasion and is not the addition the reader already knows, so the axioms have to be checked against the definition rather than against habit.

CasesAn ordinary

+

still appears on the right, inside the definition, doing its usual job on real numbers. One line therefore holds both a stipulated operation and a familiar one, distinguished only by the circle.

Also writtenAny decorated symbol serves —

\boxplus

,

*

, or an explicitly named operation. The convention is that decoration means do not assume the usual rules.

Do not confuseReading

\oplus

as a fixed standard operation. It has no meaning beyond the definition given alongside it, and a different page may use the same circle for something unrelated.

Same glyph elsewhereIn logic,

\oplus

is the fixed symbol for exclusive or, with a settled truth table. Nothing carries over: here the circle announces a local definition, there it names a specific connective.

All linear algebra symbols →↑ Vector spaces

The Field of Scalars

The scalars in a vector space come from a field — a set where addition, subtraction, multiplication, and division (by nonzero elements) all work and satisfy the standard arithmetic laws. The real numbers

\mathbb{R}

and the complex numbers

\mathbb{C}

are the two fields that appear most often in linear algebra.

A vector space over

\mathbb{R}

is called a real vector space. A vector space over

\mathbb{C}

is called a complex vector space. The choice of field determines what scalars are available for multiplication, and this affects the structure of the space. For instance, every real symmetric matrix has real eigenvalues, but a general real matrix may have complex eigenvalues — a phenomenon visible only when the scalar field extends from

\mathbb{R}

to

\mathbb{C}

.

On this site, the scalar field is

\mathbb{R}

unless explicitly stated otherwise. The axioms and definitions carry over to

\mathbb{C}

without modification.

The Standard Example: Rⁿ

The most concrete vector space is

\mathbb{R}^n

, the set of all ordered

n

-tuples of real numbers:

\mathbb{R}^n = \{(v_1, v_2, \dots, v_n) : v_i \in \mathbb{R}\}

Addition and scalar multiplication are defined entry by entry:

(u_1, \dots, u_n) + (v_1, \dots, v_n) = (u_1 + v_1, \dots, u_n + v_n)

c(v_1, \dots, v_n) = (cv_1, \dots, cv_n)

All ten axioms hold. Closure is immediate — sums and scalar products of

n

-tuples are

n

-tuples. Commutativity and associativity of vector addition follow from commutativity and associativity of real number addition applied to each component. The zero vector is

(0, 0, \dots, 0)

, and the additive inverse of

(v_1, \dots, v_n)

is

(-v_1, \dots, -v_n)

. The scalar multiplication axioms all reduce to properties of real number arithmetic applied entry by entry.

This is the vector space that underlies coordinate geometry, matrix algebra, and nearly every computational method in linear algebra. Every finite-dimensional real vector space is isomorphic to

\mathbb{R}^n

for some

n

.

Polynomial Spaces

The set

\mathcal{P}_n

of all polynomials of degree at most

n

is a vector space under ordinary polynomial addition and scalar multiplication. A typical element is

a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n

with real coefficients.

Addition combines like terms:

(a_0 + a_1 x) + (b_0 + b_1 x) = (a_0 + b_0) + (a_1 + b_1)x

. Scalar multiplication scales every coefficient:

c(a_0 + a_1 x) = ca_0 + ca_1 x

. The zero vector is the zero polynomial (all coefficients zero). The additive inverse of

p(x) = a_0 + a_1 x + \cdots + a_n x^n

is

-p(x) = -a_0 - a_1 x - \cdots - a_n x^n

.

Closure under addition holds because adding two polynomials of degree at most

n

produces a polynomial of degree at most

n

— the degree cannot increase beyond

n

. All other axioms follow from the corresponding properties of real number arithmetic applied to coefficients. The space has dimension

n + 1

, with the monomial basis

\{1, x, x^2, \dots, x^n\}

.

The set

\mathcal{P}

of all polynomials (with no degree restriction) is also a vector space, but it is infinite-dimensional: no finite set of polynomials can span it, because any finite set has a maximum degree that limits which polynomials are reachable.

Matrix Spaces

The set

\mathbb{R}^{m \times n}

of all

m \times n

real matrices is a vector space with entry-by-entry operations. Addition adds corresponding entries, and scalar multiplication scales every entry by the same scalar.

The zero vector is the

m \times n

zero matrix

O

. The additive inverse of

A = (a_{ij})

is

-A = (-a_{ij})

. All ten axioms reduce to the corresponding properties of real number arithmetic applied to each of the

mn

entries independently.

This space has dimension

mn

. The standard basis consists of the

mn

matrix units

E_{ij}

, each with a single

1

in position

(i,j)

and zeros elsewhere. Every matrix is a unique linear combination of these basis elements, with the matrix entries as coefficients.

The fact that matrices form a vector space means that the concepts of linear independence, span, and basis apply to sets of matrices — not just to column vectors. For example, the set

\{I, A, A^2\}

might be independent or dependent in

\mathbb{R}^{n \times n}

, depending on the specific matrix

A

, and answering this question uses exactly the same abstract framework as for vectors in

\mathbb{R}^n

.

Function Spaces

The set

C[a, b]

of all continuous real-valued functions on the interval

[a, b]

is a vector space with pointwise operations:

(f + g)(x) = f(x) + g(x), \qquad (cf)(x) = c \cdot f(x)

The zero vector is the function that is identically zero:

z(x) = 0

for all

x \in [a, b]

. The additive inverse of

f

is

-f

, defined by

(-f)(x) = -f(x)

.

The axioms hold because the sum of two continuous functions is continuous (closure), real-number addition is commutative and associative (so pointwise addition inherits these properties), and the distributive laws follow from ordinary scalar arithmetic applied at each point

x

.

This space is infinite-dimensional. A more structured example is the solution space of a homogeneous linear ordinary differential equation. The set of all solutions to

y'' + py' + qy = 0

(with continuous coefficients

p, q

) forms a vector space of dimension

2

: the superposition principle guarantees that any linear combination of solutions is again a solution, and the existence-uniqueness theorem guarantees that two independent solutions suffice to generate every solution.

Non-Examples

The axioms are genuine constraints, not automatic properties. Several natural-looking sets fail them.

The set of polynomials of degree exactly

n

is not a vector space. Adding two polynomials of degree

n

can cancel the leading terms — for instance,

(x^2 + x) + (-x^2 + 3) = x + 3

, which has degree

1

, not

2

. Closure under addition fails.

The set of positive real numbers with ordinary addition is not a vector space. There is no zero element: no positive real number

z

satisfies

x + z = x

for all positive

x

.

The set

\mathbb{R}^2

can be equipped with non-standard operations that violate the axioms. Defining "addition" by

(u_1, u_2) \oplus (v_1, v_2) = (u_1 + v_1, 0)

fails to produce a vector space — the second component is always destroyed, and the distributive law

c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}

breaks.

A line in

\mathbb{R}^2

that does not pass through the origin is not a subspace: it does not contain

\mathbf{0}

, and adding two points on the line generally produces a point not on the line. These failures are useful — they confirm that the axioms distinguish genuine vector spaces from imposters.

Candidate set & operations	Axiom(s) that fail	Why
Polynomials of degree exactly n; usual addition and scaling	closure under addition (axiom 1)	leading terms can cancel: (x² + x) + (−x² + 3) = x + 3, which has degree 1
Positive real numbers; ordinary addition	zero vector (axiom 4)	no positive z satisfies x + z = x for all positive x
ℝ² with (u₁, u₂) ⊕ (v₁, v₂) = (u₁ + v₁, 0)	distributivity over vector addition (axiom 8)	the second component is always destroyed; c(u + v) ≠ cu + cv in general
A line in ℝ² that does not pass through the origin; usual operations	zero vector (axiom 4) and closure under addition (axiom 1)	the line excludes 0, and the sum of two points on the line generally falls off the line

Immediate Consequences of the Axioms

Several useful facts follow from the ten axioms alone. They are not additional assumptions but provable theorems.

The scalar zero and the zero vector both annihilate under scalar multiplication:

Scalar-Zero Property

0\mathbf{v} = \mathbf{0}, \quad c\mathbf{0} = \mathbf{0}

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The proof of the first uses distributivity:

0\mathbf{v} = (0 + 0)\mathbf{v} = 0\mathbf{v} + 0\mathbf{v}

, and adding

-(0\mathbf{v})

to both sides gives

\mathbf{0} = 0\mathbf{v}

. The second is similar:

c\mathbf{0} = c(\mathbf{0} + \mathbf{0}) = c\mathbf{0} + c\mathbf{0}

.

Scaling by

-1

produces the additive inverse:

Negative One Scalar

(-1)\mathbf{v} = -\mathbf{v}

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This follows from

\mathbf{v} + (-1)\mathbf{v} = 1\mathbf{v} + (-1)\mathbf{v} = (1 + (-1))\mathbf{v} = 0\mathbf{v} = \mathbf{0}

.

If

c\mathbf{v} = \mathbf{0}

, then

c = 0

or

\mathbf{v} = \mathbf{0}

. A nonzero scalar cannot annihilate a nonzero vector — there are no zero divisors in a vector space.

The zero vector is unique, and the additive inverse of each vector is unique. Both proofs are short exercises from the axioms. These facts ensure that the algebraic structure is well-defined and free of ambiguity.

Consequence	Statement	Proof sketch
Zero scalar annihilates	0v = 0	0v = (0 + 0)v = 0v + 0v; add −(0v) to both sides
Zero vector annihilates	c0 = 0	c0 = c(0 + 0) = c0 + c0; cancel one copy
Negation by −1	(−1)v = −v	v + (−1)v = 1v + (−1)v = (1 + (−1))v = 0v = 0
No zero divisors	cv = 0 ⟹ c = 0 or v = 0	if c ≠ 0, multiply both sides by 1/c to get v = 0
Uniqueness	the zero vector and each additive inverse are unique	assume two candidates exist; the axioms force them to coincide

Why Axioms Matter

Working from axioms rather than from specific examples is the mechanism that makes linear algebra so broadly applicable.

Every theorem about linear independence applies to vectors in

\mathbb{R}^n

, to polynomials in

\mathcal{P}_n

, to matrices in

\mathbb{R}^{m \times n}

, and to functions in

C[a, b]

. Every theorem about span and basis applies identically in all these settings. The rank-nullity theorem, the theory of subspaces, the classification by dimension — none of these need to be reproved when the objects change from column vectors to polynomials.

The axioms also make clear what is not a vector space. Attempting to apply basis theory, dimension counting, or rank arguments to a set that fails the axioms produces nonsense. Checking the axioms first is a prerequisite for using any of the tools of linear algebra.

The ten axioms are not arbitrary — they are the minimal set of conditions that support the concepts of independence, span, and basis. Every axiom is used in at least one proof along the way from the definition of a vector space to the classification theorem that says two spaces are isomorphic if and only if they have the same dimension.

Summary: How the Axioms Instantiate

The ten axioms describe an abstract structure; the canonical examples make that structure concrete. The table below collects the four standard vector spaces side by side, naming what the role of "vector" is in each, the addition and scaling rules that satisfy the axioms, the specific element that serves as the zero vector, and the recipe for the additive inverse. Reading across each row instantiates the abstract axioms in a single specific space; reading down each column shows the same axiomatic ingredient taking different concrete forms.

Vector space	What "vectors" are	Operations	Zero vector	Additive inverse of v
ℝⁿ	ordered n-tuples (v₁, …, vₙ)	entry-by-entry addition and scaling	(0, 0, …, 0)	(−v₁, …, −vₙ)
𝒫ₙ (polynomials of degree ≤ n)	a₀ + a₁x + ⋯ + aₙxⁿ	combine like terms; scale every coefficient	the zero polynomial (all coefficients 0)	negate every coefficient
ℝᵐˣⁿ (real m × n matrices)	m × n grids of real entries	entry-by-entry addition and scaling	the zero matrix O (every entry 0)	negate every entry
C[a, b] (continuous functions on [a, b])	continuous functions f : [a, b] → ℝ	pointwise: (f + g)(x) = f(x) + g(x), (cf)(x) = c · f(x)	the function z(x) = 0 for all x	(−f)(x) = −f(x)

Vector Space Axioms FAQ

Is $0\mathbf{v} = \mathbf{0}$ one of the axioms?

+

No, it is a theorem proved from the ten. The proof hinges on the two zeros being different objects: the scalar zero on the left and the zero vector on the right. Writing

(0 + 0)\mathbf{v} = 0\mathbf{v} + 0\mathbf{v}

uses field addition inside the bracket and space addition outside it, and without that distinction the argument says nothing.Read more →

Why do the distributive axioms bother with parentheses?

+

Because the two plus signs are different operations. In

(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}

, the left one adds scalars in the field and the right one adds vectors in the space. The parentheses record which runs first, and the axiom exists precisely to assert that the two operations line up rather than assuming it.Read more →

Does defining addition and scaling on a set make it a vector space?

+

Not on its own. Supplying the operations is only the first step, since closure still has to be verified: the result of adding or scaling must land back inside the set. That is why continuity is a genuine hypothesis for

C[a,b]

rather than decoration, and why the non-examples fail despite having perfectly sensible operations.Read more →

Axioms related to Vector Spaces

The Ten Rules That Define the Structure

The Idea of Abstraction

The Ten Axioms

Addition Axioms

Scalar Multiplication Axioms

The ten axioms

Vector Space Axiom Notation

Vector Space Axiom Notation

The Field of Scalars

The Standard Example: Rⁿ

Polynomial Spaces

Matrix Spaces

Function Spaces

Non-Examples

Immediate Consequences of the Axioms

Why Axioms Matter

Summary: How the Axioms Instantiate

Vector Space Axioms FAQ

Is $0\mathbf{v} = \mathbf{0}$ one of the axioms?

Why do the distributive axioms bother with parentheses?

Does defining addition and scaling on a set make it a vector space?

Axioms related to Vector Spaces

The Ten Rules That Define the Structure

The Idea of Abstraction

The Ten Axioms

Addition Axioms

Scalar Multiplication Axioms

The ten axioms

Vector Space Axiom Notation

Vector Space Axiom Notation

The Field of Scalars

The Standard Example: Rⁿ

Polynomial Spaces

Matrix Spaces

Function Spaces

Non-Examples

Immediate Consequences of the Axioms

Why Axioms Matter

Summary: How the Axioms Instantiate

Vector Space Axioms FAQ

Is 0v=00\mathbf{v} = \mathbf{0}0v=0 one of the axioms?

Why do the distributive axioms bother with parentheses?

Does defining addition and scaling on a set make it a vector space?

Is $0\mathbf{v} = \mathbf{0}$ one of the axioms?