definition of quotient vectors space of vectors space by sub-'vectors space'
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of quotient set.
Target Context
- The reader will have a definition of quotient vectors space of vectors space by sub-'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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V'\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( V\): \(\in \{\text{ the sub-'vectors space's of } V'\}\)
\( \sim\): \(\in \{\text{ the equivalence relations on } V'\}\), such that \(\forall v'_1, v'_2 \in V' (v'_1 \sim v'_2 \iff v'_1 - v'_2 \in V)\)
\(*V' / V\): \(= V' / \sim\), \(\in \{\text{ the } F \text{ vectors spaces }\}\), with the operations specified below
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Conditions:
\(\forall r \in F, \forall [v_1], [v_2] \in V' / V (r [v_1] = [r v_1] \land [v_1] + [v_2] = [v_1 + v_2])\)
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2: Note
Let us see that \(V' / V\) is indeed well-defined.
Let us see that \(\sim\) is indeed an equivalence relation on \(V'\).
For each \(v' \in V'\), \(v' \sim v'\), because \(v' - v' = 0 \in V\).
For each \(v'_1, v'_2 \in V'\), \(v'_1 \sim v'_2\) implies that \(v'_2 \sim v'_1\), because as \(v'_1 - v'_2 \in V\), \(v'_2 - v'_1 = - (v'_1 - v'_2) \in V\).
For each \(v'_1, v'_2, v'_3 \in V'\), \(v'_1 \sim v'_2\) and \(v'_2 \sim v'_3\) implies that \(v'_1 \sim v'_3\), because as \(v'_1 - v'_2 \in V\) and \(v'_2 - v'_3 \in V\), \(v'_1 - v'_3 = v'_1 - v'_2 + v'_2 - v'_3 = (v'_1 - v'_2) + (v'_2 - v'_3) \in V\).
So, \(V' / \sim\) 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 \([r w_1] = [r (v_1 + v)] = [r v_1 + r v]\), but \(r v_1 + r v - r v_1 = r v \in V\), 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 \(V' / V\) is indeed an \(F\) vectors space.
1) for any elements, \([v_1], [v_2] \in V' / V\), \([v_1] + [v_2] \in V' / V\) (closed-ness under addition): \([v_1] + [v_2] = [v_1 + v_2] \in V' / V\).
2) for any elements, \([v_1], [v_2] \in V' / V\), \([v_1] + [v_2] = [v_2] + [v_1]\) (commutativity of addition): \([v_1] + [v_2] = [v_1 + v_2] = [v_2 + v_1] = [v_2] + [v_1]\).
3) for any elements, \([v_1], [v_2], [v_3] \in V' / V\), \(([v_1] + [v_2]) + [v_3] = [v_1] + ([v_2] + [v_3])\) (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, \([0] \in V' / V\), such that for any \([v] \in V' / V\), \([v] + [0] = [v]\) (existence of 0 vector): \([0] \in V' / V\), and \([v] + [0] = [v + 0] = [v]\).
5) for any element, \([v] \in V' / V\), there is an inverse element, \([v'] \in V' / V\), such that \([v'] + [v] = [0]\) (existence of inverse vector): \([v'] := [- v] \in V' / V\), and \([v'] + [v] = [- v] + [v] = [-v + v] = [0]\).
6) for any element, \([v] \in V' / V\), and any scalar, \(r \in F\), \(r . [v] \in V' / V\) (closed-ness under scalar multiplication): \(r . [v] = [r v] \in V' / V\).
7) for any element, \([v] \in V' / V\), 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, \([v_1], [v_2] \in V' / V\), 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, \([v] \in V' / V\), 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, \([v] \in V' / V\), \(1 . [v] = [v]\) (identity of 1 multiplication): \(1 . [v] = [1 v] = [v]\).
