A Note on Vector Space Axioms
- PDF / 142,924 Bytes
- 12 Pages / 595 x 842 pts (A4) Page_size
- 87 Downloads / 184 Views
A Note on Vector Space Axioms∗ Aniruddha V Deshmukh
In this article, we will see why all the axioms of a vector space are important in its definition. During a regular course, when an undergraduate student encounters the definition of vector spaces for the first time, it is natural for the student to think of some axioms as redundant and unnecessary. In this article, we shall deal with only one axiom 1 · v = v and its importance. In the article, we would first try to prove that it is redundant just as an undergraduate student would (in the first attempt), and then point out the mistake in the proof, and provide an example which will be sufficient to show the importance of the axiom.
Aniruddha V Deshmukh has completed his masters from S V National Institute of Technology, Surat and is the gold medalist of his batch. His interests lie in linear algebra, metric spaces,
1. Definitions and Preliminaries
topology, functional analysis,
All the definitions are taken directly from [1]
and differential geometry. He is also keen on teaching
Definition 1 (Field). Let F be a non-empty set. Define two operations + : F × F → F and · : F × F → F. Eventually, these operations will be called ‘addition’ and ‘multiplication’. Clearly, both are binary operations. Now, (F, +, ·) is a field if 1. ∀x, y, z ∈ F, x + (y + z) = (x + y) + z (Associativity of addition)
mathematics.
Keywords Vector spaces, fields, axioms,
2. ∃0 ∈ F such that ∀x ∈ F, 0 + x = x + 0 = x (Existence of additive identity)
Zorn’s lemma.
3. ∀x ∈ F, ∃y ∈ F such that x + y = y + x = 0 (Existence of additive inverses for every element) 4. ∀x, y ∈ F, x + y = y + x (Commutativity of addition)
∗
Vol.25, No.11, DOI: https://doi.org/10.1007/s12045-020-1074-z
RESONANCE | November 2020
1547
GENERAL ARTICLE
5. ∀x, y, z ∈ F, x · (y · z) = (x · y) · z (Associativity of multiplication) 6. ∃1 ∈ F such that ∀x ∈ F, 1 · x = x · 1 = x (Existence of multiplicative identity, called the ‘unity’) 7. ∀x , 0 ∈ F, ∃y ∈ F such that x · y = y · x = 1 (Existence of multiplicative inverses for every element other than additive identity) 8. ∀x, y ∈ F, x · y = y · x (Commutativity of multiplication) 9. ∀x, y, z ∈ F, x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z (Multiplication is distributive over addition)
Remark. We shall denote the additive inverse of x ∈ F by −x and 1 the multiplicative inverse of x ∈ F, where x , 0, by . x Since later, ‘scalar multiplication’ will be defined for a vector space, we will not use ‘·’ for multiplication of two elements of a field. Rather, if α and β are two elements of a field F, their multiplication will be shown by αβ rather than α · β to avoid confusion. Definition 2 (Vector Space). Let V be a non-empty set and F be a field. Define two operations + : V × V → V and · : F × V → V which will be eventually called the ‘vector addition’ and ‘scalar multiplication’ respectively. Clearly, + is a binary operation. Now, (V, +, ·) is a vector space over F if 1. ∀u, v, w ∈ V, (u + v) + w = u + (v + w) (Associativity of addition) 2. ∃0 ∈ V s
Data Loading...
