definition of product vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description 1
- 2: Structured Description 2
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of product set.
Target Context
- The reader will have a definition of product 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 1
Here is the rules of Structured Description.
Entities:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_j \vert j \in J\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\(*\times_{j \in J} V_j\): \(= \text{ the product set }\), \(\in \{\text{ the } F \text{ vectors spaces }\}\), with the operations specified below
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Conditions:
\(\forall r \in F, \forall f \in \times_{j \in J} V_j (\forall j \in J ((r f) (j) = r (f (j))))\)
\(\land\)
\(\forall f, f' \in \times_{j \in J} V_j (\forall j \in J ((f + f') (j) = f (j) + f' (j)))\)
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2: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_1, ..., V_k\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\(*V_1 \times ... \times V_k\): \(= \text{ the product set }\), \(\in \{\text{ the } F \text{ vectors spaces }\}\), with the operations specified below
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Conditions:
\(\forall r \in F, \forall v = (v_1, ..., v_k) \in V_1 \times ... \times V_k (r v = (r v_1, ..., r v_k))\)
\(\land\)
\(\forall v = (v_1, ..., v_k), v' = (v'_1, ..., v'_k) \in V_1 \times ... \times V_k (v + v' = (v_1 + v'_1, ..., v_k + v'_k))\)
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3: Note
Let us see that \(\times_{j \in J} V_j\) is indeed an \(F\) vectors space.
1) for any elements, \(f, f' \in \times_{j \in J} V_j\), \(f + f' \in \times_{j \in J} V_j\) (closed-ness under addition): for each \(j \in J\), \((f + f') (j) = f (j) + f' (j) \in V_j\).
2) for any elements, \(f, f' \in \times_{j \in J} V_j\), \(f + f' = f' + f\) (commutativity of addition): for each \(j \in J\), \((f + f') (j) = f (j) + f' (j) = f' (j) + f (j) = (f' + f) (j)\).
3) for any elements, \(f, f', f'' \in \times_{j \in J} V_j\), \((f + f') + f'' = f + (f' + f'')\) (associativity of additions): for each \(j \in J\), \(((f + f') + f'') (j) = (f + f') (j) + f'' (j) = f (j) + f' (j) + f'' (j) = f (j) + (f' + f'') (j) = (f + (f' + f'')) (j)\).
4) there is a 0 element, \(0 \in \times_{j \in J} V_j\), such that for any \(f \in \times_{j \in J} V_j\), \(f + 0 = f\) (existence of 0 vector): \(f_0 \in \times_{j \in J} V_j\) such that for each \(j \in J\), \(f_0 (j) = 0\), is \(0\), because \((f + f_0) (j) = f (j) + f_0 (j) = f (j)\).
5) for any element, \(f \in \times_{j \in J} V_j\), there is an inverse element, \(f' \in \times_{j \in J} V_j\), such that \(f' + f = 0\) (existence of inverse vector): \(f' \in \times_{j \in J} V_j\) such that for each \(j \in J\), \(f' (j) = - f (j)\), is such a one, because \((f' + f) (j) = f' (j) + f (j) = - f (j) + f (j) = 0\).
6) for any element, \(f \in \times_{j \in J} V_j\), and any scalar, \(r \in F\), \(r . f \in \times_{j \in J} V_j\) (closed-ness under scalar multiplication): \((r . f) (j) = r f (j) \in V_j\).
7) for any element, \(f \in \times_{j \in J} V_j\), and any scalars, \(r_1, r_2 \in F\), \((r_1 + r_2) . f = r_1 . f + r_2 . f\) (scalar multiplication distributability for scalars addition): for each \(j \in J\), \(((r_1 + r_2) . f) (j) = (r_1 + r_2) f (j) = r_1 f (j) + r_2 f (j) = (r_1 f) (j) + (r_2 f) (j) = (r_1 . f + r_2 . f) (j)\).
8) for any elements, \(f, f' \in \times_{j \in J} V_j\), and any scalar, \(r \in F\), \(r . (f + f') = r . f + r . f'\) (scalar multiplication distributability for vectors addition): for each \(j \in J\), \((r . (f + f')) (j) = r (f + f') (j) = r (f (j) + f' (j)) = r f (j) + r f' (j) = (r f) (j) + (r f') (j) = (r . f + r . f') (j)\).
9) for any element, \(f \in \times_{j \in J} V_j\), and any scalars, \(r_1, r_2 \in F\), \((r_1 r_2) . f = r_1 . (r_2 . f)\) (associativity of scalar multiplications): for each \(j \in J\), \(((r_1 r_2) . f) (j) = (r_1 r_2) f (j) = r_1 (r_2 f (j)) = r_1 ((r_2 f) (j)) = (r_1 . (r_2 . f)) (j)\).
10) for any element, \(f \in \times_{j \in J} V_j\), \(1 . f = f\) (identity of 1 multiplication): for each \(j \in J\), \((1 . f) (j) = 1 . (f (j)) = f (j)\).
Let us see that \(V_1 \times ... \times V_k\) is indeed an \(F\) vectors space.
1) for any elements, \(v, v' \in V_1 \times ... \times V_k\), \(v + v' \in V_1 \times ... \times V_k\) (closed-ness under addition): for \(v = (v_1, ..., v_k)\) and \(v' = (v'_1, ..., v'_k)\), \(v + v' = (v_1 + v'_1, ..., v_k + v'_k) \in V_1 \times ... \times V_k\).
2) for any elements, \(v, v' \in V_1 \times ... \times V_k\), \(v + v' = v' + v\) (commutativity of addition): for \(v = (v_1, ..., v_k)\) and \(v' = (v'_1, ..., v'_k)\), \(v + v' = (v_1, ..., v_k) + (v'_1, ..., v'_k) = (v_1 + v'_1, ..., v_k + v'_k) = (v'_1 + v_1, ..., v'_k + v_k) = (v'_1, ..., v'_k) + (v_1, ..., v_k) = v' + v\).
3) for any elements, \(v, v', v'' \in V_1 \times ... \times V_k\), \((v + v') + v'' = v + (v' + v'')\) (associativity of additions): for \(v = (v_1, ..., v_k)\), \(v' = (v'_1, ..., v'_k)\), and \(v'' = (v''_1, ..., v''_k)\), \((v + v') + v'' = (v_1 + v'_1, ..., v_k + v'_k) + (v''_1, ..., v''_k) = (v_1 + v'_1 + v''_1, ..., v_k + v'_k + v''_k) = (v_1, ..., v_k) + (v'_1 + v''_1, ..., v'_k + v''_k) = v + (v' + v'')\).
4) there is a 0 element, \(0 \in V_1 \times ... \times V_k\), such that for any \(v \in V_1 \times ... \times V_k\), \(v + 0 = v\) (existence of 0 vector): \(v_0 = (0, ..., 0) \in V_1 \times ... \times V_k\) is \(0\), because for \(v = (v_1, ..., v_k)\), \(v_0 + v = (0 + v_1, ..., 0 + v_k) = (v_1, ..., v_k) = v\).
5) for any element, \(v \in V_1 \times ... \times V_k\), there is an inverse element, \(v' \in V_1 \times ... \times V_k\), such that \(v' + v = 0\) (existence of inverse vector): for \(v = (v_1, ..., v_k)\), \(v' = (- v_1, ..., - v_k) \in V_1 \times ... \times V_k\) is a one, because \(v' + v = (- v_1, ..., - v_k) + (v_1, ..., v_k) = (- v_1 + v_1, ..., - v_k + v_k) = (0, ..., 0) = 0\).
6) for any element, \(v \in V_1 \times ... \times V_k\), and any scalar, \(r \in F\), \(r . v \in V_1 \times ... \times V_k\) (closed-ness under scalar multiplication): for \(v = (v_1, ..., v_k)\), \(r . v = (r v_1, ..., r v_k) \in V_1 \times ... \times V_k\).
7) for any element, \(v \in V_1 \times ... \times V_k\), 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): for \(v = (v_1, ..., v_k)\), \((r_1 + r_2) . v = ((r_1 + r_2) v_1, ..., (r_1 + r_2) v_k) = (r_1 v_1 + r_2 v_1, ..., r_1 v_k + r_2 v_k) = (r_1 v_1, ..., r_1 v_k) + (r_2 v_1, ..., r_2 v_k) = r_1 (v_1, ..., v_k) + r_2 (v_1, ..., v_k) = r_1 . v_1 + r_2 . v_2\).
8) for any elements, \(v, v' \in V_1 \times ... \times V_k\), and any scalar, \(r \in F\), \(r . (v + v') = r . v + r . v'\) (scalar multiplication distributability for vectors addition): for \(v = (v_1, ..., v_k)\) and \(v' = (v'_1, ..., v'_k)\), \(r . (v + v') = r (v_1 + v'_1, ..., v_k + v'_k) = (r (v_1 + v'_1), ..., r (v_k + v'_k)) = (r v_1 + r v'_1, ..., r v_k + r v'_k) = (r v_1, ..., r v_k) + (r v'_1, ..., r v'_k) = r (v_1, ..., v_k) + r (v'_1, ..., v'_k) = r . v + r . v'\).
9) for any element, \(v \in V_1 \times ... \times V_k\), and any scalars, \(r_1, r_2 \in F\), \((r_1 r_2) . v = r_1 . (r_2 . v)\) (associativity of scalar multiplications): for \(v = (v_1, ..., v_k)\), \((r_1 r_2) . v = ((r_1 r_2) v_1, ..., (r_1 r_2) v_k) = (r_1 (r_2 v_1), ..., r_1 (r_2 v_k)) = r_1 (r_2 v_1, ..., r_2 v_k) = r_1 . (r_2 . v)\).
10) for any element, \(v \in V_1 \times ... \times V_k\), \(1 . v = v\) (identity of 1 multiplication): for \(v = (v_1, ..., v_k)\), \(1 . v = (1 v_1, ..., 1 v_k) = (v_1, ..., v_k) = v\).
