description/proof of that range of linear map between vectors spaces is sub-'vectors space' of codomain
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 linear map.
Target Context
- The reader will have a description and a proof of the proposition that the range of any linear map between any vectors spaces is a sub-'vectors space' of the codomain.
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:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
Statements:
\(f (V_1) \in \{\text{ the sub-'vectors space's of } V_2\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(f (V_1)\) satisfies the requirements to be a vectors space.
Step 1:
1) for any elements, \(f (v), f (v') \in f (V_1)\), \(f (v) + f (v') \in f (V_1)\) (closed-ness under addition): \(f (v) + f (v') = f (v + v') \in f (V_1)\).
2) for any elements, \(f (v), f (v') \in f (V_1)\), \(f (v) + f (v') = f (v') + f (v)\) (commutativity of addition): it holds on ambient \(V_2\).
3) for any elements, \(f (v), f (v'), f (v'') \in f (V_1)\), \((f (v) + f (v')) + f (v'') = f (v) + (f (v') + f (v''))\) (associativity of additions): it holds on ambient \(V_2\).
4) there is a 0 element, \(0 \in f (V_1)\), such that for any \(f (v) \in f (V_1)\), \(f (v) + 0 = f (v)\) (existence of 0 vector): \(f (0) = 0 \in f (V_1)\) is the one.
5) for any element, \(f (v) \in f (V_1)\), there is an inverse element, \(f (v') \in f (V_1)\), such that \(f (v') + f (v) = 0\) (existence of inverse vector): \(f (- v) = - f (v) \in f (V_1)\) is the one.
6) for any element, \(f (v) \in f (V_1)\), and any scalar, \(r \in F\), \(r . f (v) \in f (V_1)\) (closed-ness under scalar multiplication): \(f (r v) = r . f (v) \in f (V_1)\).
7) for any element, \(f (v) \in f (V_1)\), and any scalars, \(r_1, r_2 \in F\), \((r_1 + r_2) . f (v) = r_1 . f (v) + r_2 . f (v)\) (scalar multiplication distributability for scalars addition): it holds on ambient \(V_2\).
8) for any elements, \(f (v), f (v') \in f (V_1)\), and any scalar, \(r \in F\), \(r . (f (v) + f (v')) = r . f (v) + r . f (v')\) (scalar multiplication distributability for vectors addition): it holds on ambient \(V_2\).
9) for any element, \(f (v) \in f (V_1)\), and any scalars, \(r_1, r_2 \in F\), \((r_1 r_2) . f (v) = r_1 . (r_2 . f (v))\) (associativity of scalar multiplications): it holds on ambient \(V_2\).
10) for any element, \(f (v) \in f (V_1)\), \(1 . f (v) = f (v)\) (identity of 1 multiplication): it holds on ambient \(V_2\).
