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MTH6140 Linear Algebra II: Fields and Vector Spaces

This document defines a field and vector space. A field is an algebraic structure where addition and multiplication satisfy certain laws, such as having unique sums/products and inverses. Examples of fields include rational, real, and complex numbers. A vector space over a field consists of elements that can be added together and multiplied by scalars, obeying laws like unique sums/products and distributing scalars over addition. The set of n-tuples over a field forms an important example of a vector space.

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55 views2 pages

MTH6140 Linear Algebra II: Fields and Vector Spaces

This document defines a field and vector space. A field is an algebraic structure where addition and multiplication satisfy certain laws, such as having unique sums/products and inverses. Examples of fields include rational, real, and complex numbers. A vector space over a field consists of elements that can be added together and multiplied by scalars, obeying laws like unique sums/products and distributing scalars over addition. The set of n-tuples over a field forms an important example of a vector space.

Uploaded by

Roy Vesey
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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MTH6140 Linear Algebra II

Fields and vector spaces Summary

A field is an algebraic structure K in which we can add and multiply elements,


such that the following laws hold:
Addition laws

(FA0) For any a, b ∈ K, there is a unique element a + b ∈ K.


(FA1) For all a, b, c ∈ K, we have a + (b + c) = (a + b) + c.
(FA2) There is an element 0 ∈ K such that a + 0 = 0 + a = a for all a ∈ K.
(FA3) For any a ∈ K, there exists −a ∈ K such that a + (−a) = (−a) + a = 0.
(FA4) For any a, b ∈ K, we have a + b = b + a.

Multiplication laws

(FM0) For any a, b ∈ K, there is a unique element ab ∈ K.


(FM1) For all a, b, c ∈ K, we have a(bc) = (ab)c.
(FM2) There is an element 1 ∈ K such that a1 = 1a = a for all a ∈ K.
(FM3) For any a ∈ K with a 6= 0, there exists a−1 ∈ K such that aa−1 = a−1 a = 1.
(FM4) For any a, b ∈ K, we have ab = ba.

Distributive law

(D) For all a, b, c ∈ K, we have a(b + c) = ab + ac.

Note the similarity of the addition and multiplication laws. We say that (K, +) is
an abelian group if (FA0)–(FA4) hold. Then (FM0)–(FM4) say that (K \ {0}, ·) is also
an abelian group. (We have to leave out 0 because, as (FM3) says, 0 does not have a
multiplicative inverse.)
Examples of fields include Q (the rational numbers), R (the real numbers), C (the
complex numbers), and F p (the integers mod p, for p a prime number).

1
Let K be a field. A vector space V over K is an algebraic structure in which we
can add two elements of V , and multiply an element of V by an element of K (this is
called scalar multiplication), such that the following rules hold:

Addition laws

(VA0) For any u, v ∈ V , there is a unique element u + v ∈ V .


(VA1) For all u, v, w ∈ V , we have u + (v + w) = (u + v) + w.
(VA2) There is an element 0 ∈ V such that v + 0 = 0 + v = av for all v ∈ V .
(VA3) For any v ∈ V , there exists −v ∈ V such that v + (−v) = (−v) + v = 0.
(VA4) For any u, v ∈ K, we have u + v = v + u.

Scalar multiplication laws

(VM0) For any a ∈ K, v ∈ V , there is a unique element av ∈ V .


(VM1) For any a ∈ K, u, v ∈ V , we have a(u + v) = au + av.
(VM2) For any a, b ∈ K, v ∈ V , we have (a + b)v = av + bv.
(VM3) For any a, b ∈ K, v ∈ V , we have (ab)v = a(bv).
(VM4) For any v ∈ V , we have 1v = v (where 1 is the element given by (FM2)).

Again, we can summarise (VA0)–(VA4) by saying that (V, +) is an abelian group.


The most important example of a vector space over a field K is the set K n of all
n-tuples of elements of K: the addition and scalar multiplication are defined by the
rules

(u1 , u2 , . . . , un ) + (v1 , v2 , . . . , vn ) = (u1 + v1 , u2 + v2 , . . . , un + vn ),


a(v1 , v2 , . . . , vn ) = (av1 , av2 , . . . , avn ).

You may assume this information, and are not expected to provide proofs of what
is claimed here.

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