1020: Quotient Vectors Space of Vectors Space by Sub-'Vectors Space'

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definition of quotient vectors space of vectors space by sub-'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: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of quotient set.

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Target Context

• The reader will have a definition of quotient vectors space of vectors space by sub-'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: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V^{\prime }$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$V$: $\in \{\text{ the sub-'vectors space's of }{V}^{\prime }\}$
$\sim$: $\in \{\text{ the equivalence relations on }{V}^{\prime }\}$, such that $\mathrm{\forall }{v}_1^{\prime },v_2^{\prime }\in V^{\prime }(v_1^{\prime }\sim v_2^{\prime }⟺v_1^{\prime }-v_2^{\prime }\in V)$
$\ast V^{\prime }/V$: $=V^{\prime }/\sim$, $\in \{\text{ the }F\text{ vectors spaces }\}$, with the operations specified below
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Conditions:
$\mathrm{\forall }r\in F,\mathrm{\forall }[v_1],[v_2]\in V^{\prime }/V(r[v_1]=[r{v}_1]\wedge [v_1]+[v_2]=[v_1+v_2])$
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2: Note

Let us see that $V^{\prime }/V$ is indeed well-defined.

Let us see that $\sim$ is indeed an equivalence relation on $V^{\prime }$.

For each $v^{\prime }\in V^{\prime }$, $v^{\prime }\sim v^{\prime }$, because $v^{\prime }-v^{\prime }=0\in V$.

For each $v_1^{\prime },v_2^{\prime }\in V^{\prime }$, $v_1^{\prime }\sim v_2^{\prime }$ implies that $v_2^{\prime }\sim v_1^{\prime }$, because as $v_1^{\prime }-v_2^{\prime }\in V$, $v_2^{\prime }-v_1^{\prime }=-(v_1^{\prime }-v_2^{\prime })\in V$.

For each $v_1^{\prime },v_2^{\prime },v_3^{\prime }\in V^{\prime }$, $v_1^{\prime }\sim v_2^{\prime }$ and $v_2^{\prime }\sim v_3^{\prime }$ implies that $v_1^{\prime }\sim v_3^{\prime }$, because as $v_1^{\prime }-v_2^{\prime }\in V$ and $v_2^{\prime }-v_3^{\prime }\in V$, $v_1^{\prime }-v_3^{\prime }=v_1^{\prime }-v_2^{\prime }+v_2^{\prime }-v_3^{\prime }=(v_1^{\prime }-v_2^{\prime })+(v_2^{\prime }-v_3^{\prime })\in V$.

So, $V^{\prime }/\sim$ is well-defined as a set.

Let us see that the operations are well-defined.

For $r[v_1]=[r{v}_1]$, $[r{v}_1]$ does not depend on the choice of the representative, $v_1$, because letting $[v_1]=[w_1]$, $w_1-v_1:=v\in V$, and $[r{w}_1]=[r(v_1+v)]=[r{v}_1+rv]$, but $r{v}_1+rv-r{v}_1=rv\in V$, so, $[r{w}_1]=[r{v}_1]$.

For $[v_1]+[v_2]=[v_1+v_2]$, $[v_1+v_2]$ does not depend on the choices of the representatives, $v_1,v_2$, because letting $[v_1]=[w_1]$ and $[v_2]=[w_2]$, as $w_1-v_1,w_2-v_2\in V$, $w_1+w_2-(v_1+v_2)=(w_1-v_1)+(w_2-v_2)\in V$, so, $[w_1+w_2]=[v_1+v_2]$.

Let us see that $V^{\prime }/V$ is indeed an $F$ vectors space.

1) for any elements, $[v_1],[v_2]\in V^{\prime }/V$, $[v_1]+[v_2]\in V^{\prime }/V$ (closed-ness under addition): $[v_1]+[v_2]=[v_1+v_2]\in V^{\prime }/V$.

2) for any elements, $[v_1],[v_2]\in V^{\prime }/V$, $[v_1]+[v_2]=[v_2]+[v_1]$ (commutativity of addition): $[v_1]+[v_2]=[v_1+v_2]=[v_2+v_1]=[v_2]+[v_1]$.

3) for any elements, $[v_1],[v_2],[v_3]\in V^{\prime }/V$, $([v_1]+[v_2])+[v_3]=[v_1]+([v_2]+[v_3])$ (associativity of additions): $([v_1]+[v_2])+[v_3]=[v_1+v_2]+[v_3]=[v_1+v_2+v_3]=[v_1]+[v_2+v_3]=[v_1]+([v_2]+[v_3])$.

4) there is a 0 element, $[0]\in V^{\prime }/V$, such that for any $[v]\in V^{\prime }/V$, $[v]+[0]=[v]$ (existence of 0 vector): $[0]\in V^{\prime }/V$, and $[v]+[0]=[v+0]=[v]$.

5) for any element, $[v]\in V^{\prime }/V$, there is an inverse element, $[v^{\prime }]\in V^{\prime }/V$, such that $[v^{\prime }]+[v]=[0]$ (existence of inverse vector): $[v^{\prime }]:=[-v]\in V^{\prime }/V$, and $[v^{\prime }]+[v]=[-v]+[v]=[-v+v]=[0]$.

6) for any element, $[v]\in V^{\prime }/V$, and any scalar, $r\in F$, $r.[v]\in V^{\prime }/V$ (closed-ness under scalar multiplication): $r.[v]=[rv]\in V^{\prime }/V$.

7) for any element, $[v]\in V^{\prime }/V$, 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): $(r_1+r_2).[v]=[(r_1+r_2)v]=[r_{1}v+r_{2}v]=[r_{1}v]+[r_{2}v]=r_1.[v]+r_2.[v]$.

8) for any elements, $[v_1],[v_2]\in V^{\prime }/V$, and any scalar, $r\in F$, $r.([v_1]+[v_2])=r.[v_1]+r.[v_2]$ (scalar multiplication distributability for vectors addition): $r.([v_1]+[v_2])=r.[v_1+v_2]=[r(v_1+v_2)]=[r{v}_1+r{v}_2]=[r{v}_1]+[r{v}_2]=r.[v_1]+r.[v_2]$.

9) for any element, $[v]\in V^{\prime }/V$, and any scalars, $r_1,r_2\in F$, $(r_1{r}_2).[v]=r_1.(r_2.[v])$ (associativity of scalar multiplications): $(r_1{r}_2).[v]=[(r_1{r}_2)v]=[r_1(r_{2}v)]=r_1.(r_2.[v])$.

10) for any element, $[v]\in V^{\prime }/V$, $1.[v]=[v]$ (identity of 1 multiplication): $1.[v]=[1v]=[v]$.

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References

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