A description/proof of that set of vectors space homomorphisms constitutes vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of vectors space.
- The reader knows a definition of %structure kind name% homomorphism.
Target Context
- The reader will have a description and a proof of the proposition that for any 2 vectors spaces over any same field, the set of vectors space homomorphisms constitutes a 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: Description
For any vectors spaces, \(V_1, V_2\), over any same field, \(F\), the set of vectors space homomorphisms, \(Hom (V_1, V_2) := \{f: V_1 \rightarrow V_2\}\), is a vectors space over \(F\) with the vectors space operation defined as \((r_1 f_1 + r_2 f_2) (v) = r_1 f_1 (v) + r_2 f_2 (v)\) where \(r_i \in F\), \(f_i \in Hom (V_1, V_2)\), and \(v \in V_1\).
2: Proof
\(r_1 f_1 (v) + r_2 f_2 (v) \in V_2\). So, \(r_1 f_1 + r_2 f_2\) is a map from \(V_1\) to \(V_2\). Is \(r_1 f_1 + r_2 f_2\) a vectors space homomorphism? For any \(r_3, r_4 \in F\) and any \(v_1, v_2 \in V_1\), \((r_1 f_1 + r_2 f_2) (r_3 v_1 + r_4 v_2) = r_1 f_1 (r_3 v_1 + r_4 v_2) + r_2 f_2 (r_3 v_1 + r_4 v_2) = r_1 r_3 f_1 (v_1) + r_1 r_4 f_1 (v_2) + r_2 r_3 f_2 (v_1) + r_2 r_4 f_2 (v_2) = r_3 (r_1 f_1 (v_1) + r_2 f_2 (v_1)) + r_4 (r_1 f_1 (v_2) + r_2 f_2 (v_2)) = r_3 ( (r_1 f_1 + r_2 f_2) (v_1)) + r_4 ( (r_1 f_1 + r_2 f_2) (v_2))\). So, yes, \(r_1 f_1 + r_2 f_2\) is a vectors space homomorphism.
