406: %field name% vectors space

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definition of %field name% vectors space

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topics

about: vectors space

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the table of contents of this article

• starting context
• target context
• orientation
• main body
• 1: definition
• 2: note

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starting context

• the reader knows a definition of field.
• the reader knows a definition of set.

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target context

• the reader will have a definition of %field name% vectors space.

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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.

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main body

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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, $v^{\prime }\in v$, such that $v^{\prime }+v=0$ (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)

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2: note

$.$ is often omitted in notations like $rv$ 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.

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references

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