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: definition
any set, \(v\), with any \(+\) (addition) operation and any \(.\) (scaler multiplication) operation with respect to any field, \(f\), that satisfies these conditions while any element of \(v\) is called 'vector': 1) for any elements, \(v_1, v_2 \in v\), \(v_1 + v_2 \in v\) (closed-ness under addition); 2) for any elements, \(v_1, v_2 \in v\), \(v_1 + v_2 = v_2 + v_1\) (commutativity of addition); 3) for any elements, \(v_1, v_2, v_3 \in v\), \((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)\) (associativity of additions); 4) there is a 0 element, \(0 \in v\), such that for any \(v \in v\), \(v + 0 = v\) (existence of 0 vector); 5) for any element, \(v \in v\), there is an inverse element, \(v' \in v\), such that \(v' + v = 0\) (existence of inverse vector); 6) for any element, \(v \in v\), and any scalar, \(r \in f\), \(r . v \in v\) (closed-ness under scalar multiplication); 7) for any element, \(v \in v\), and any scalars, \(r_1, r_2 \in f\), \((r_1 + r_2) . v = r_1 . v + r_2 . v\) (scalar multiplication distributability for scalars addition); 8) for any elements, \(v_1, v_2 \in v\), and any scalar, \(r \in f\), \(r . (v_1 + v_2) = r . v_1 + r . v_2\) (scalar multiplication distributability for vectors addition); 9) for any element, \(v \in v\), and any scalars, \(r_1, r_2 \in f\), \((r_1 r_2) . v = r_1 . (r_2 . v)\) (associativity of scalar multiplications); 10) for any element, \(v \in v\), \(1 . v = v\) (identity of 1 multiplication)
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.
