definition of %field name% vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of field.
- The reader knows a definition of set.
Target Context
- The reader will have a definition of %field name% vectors space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\(*V\): \(\in \{\text{ the sets }\}\) with any \(+: V \times V \to V\) (addition) operation and any \(.: F \times V \to V\) (scalar multiplication) operation
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Conditions:
1) \(\forall v_1, v_2 \in V (v_1 + v_2 \in V)\) (closed-ness under addition)
2) \(\forall v_1, v_2 \in V (v_1 + v_2 = v_2 + v_1)\) (commutativity of addition)
3) \(\forall v_1, v_2, v_3 \in V ((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3))\) (associativity of additions)
4) \(\exists 0 \in V (\forall v \in V (v + 0 = v))\) (existence of 0 element)
5) \(\forall v \in V (\exists v' \in V (v' + v = 0))\) (existence of inverse element)
6) \(\forall v \in V, \forall r \in F (r . v \in V)\) (closed-ness under scalar multiplication)
7) \(\forall v \in V, \forall r_1, r_2 \in F ((r_1 + r_2) . v = r_1 . v + r_2 . v)\) (scalar multiplication distributability for scalars addition)
8) \(\forall v_1, v_2 \in V, \forall r \in F (r . (v_1 + v_2) = r . v_1 + r . v_2)\) (scalar multiplication distributability for vectors addition)
9) \(\forall v \in V, \forall r_1, r_2 \in F ((r_1 r_2) . v = r_1 . (r_2 . v))\) (associativity of scalar multiplications)
10) \(\forall v \in V (1 . v = v)\) (identity of 1 multiplication)
Any element of \(V\) is called "vector".
2: Note
\(.\) is often omitted in notations like \(r v\) instead of \(r . v\).
Any 'the real numbers field' vectors space and any 'the complex numbers field' vectors space are often called real vectors space and complex vectors space, respectively.
As any field is a ring, any vectors space is a module.
Any module whose ring happens to be a field is a vectors space, because the Conditions s are practically the same, so, the module satisfies Conditions for vectors space.
