Vector space

A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.

Formal definition

A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations:

Related Topics:
Field - Real - Complex - Set

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• vector addition: V × V → V denoted v + w, where v, w ∈ V, and
• scalar multiplication: F × V → V denoted a v, where a ∈ F and v ∈ V.

which satisfy following axioms (for all a, b ∈ F and u, v, and w ∈ V):

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• Vector addition is associative: u + (v + w) = (u + v) + w.
• Vector addition is commutative: v + w = w + v.
• There exists an additive identity element 0 in V, such that for all elements v in V, v + 0 = v.
• For all v in V, there exists an element w in V, such that v + w = 0.
• Scalar multiplication is associative: a(bv) = (ab)v.
• 1 v = v, where 1 denotes the multiplicative identity in F.
• Scalar multiplication distributes over vector addition: a(v + w) = a v + a w.
• Scalar multiplication distributes over scalar addition: (a + b)v = a v + b v.

The elements of V are called vectors and the elements of F are called scalars. In most applications the field of scalars is the real or complex numbers.

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• A vector space over the field of real numbers R is called a real vector space.
• A vector space over the field of complex numbers C is called a complex vector space.

The concept of a vector space is entirely abstract, like the concepts of a group, ring, and field. To determine if a set V is a vector space, one only has to specify the set V, a field F, and define vector addition and scalar multiplication in V. Then, if V satisfies the above ten axioms, it is a vector space over the field F.

Related Topics:
Group - Ring - Field

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Elementary properties

There are a number of properties that follow easily from the vector space axioms. These include:

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• The zero vector 0 ∈ V (defined by axiom 3) is unique.
• a 0 = 0 for all a ∈ F.
• 0 v = 0 for all v ∈ V where 0 is the additive identity in F.
• a v = 0 if and only if either a = 0 or v = 0.
• The additive inverse of a vector v (defined by axiom 4) is unique. It is usually denoted −v. The notation v − w for v + (−w) is also standard.
• (−1)v = −v for all v ∈ V.
• (−a)v = a(−v) = −(av) for all a ∈ F and all v ∈ V.