description/proof of that range of linear map between vectors spaces is vectors subspace of codomain
Topics
About: vectors space
The table of contents of this article
Starting Context
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 vectors subspace 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:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
\(f (V_1)\):
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Statements:
\(f (V_1) \in \{\text{ the vectors subspaces of } V_2\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f (V_1)\) is closed under linear combination.
Step 1:
Let us see that \(f (V_1)\) is closed under linear combination.
Let \(f (v), f (v') \in f (V_1)\) and \(r, r' \in F\) be any.
\(r f (v) + r' f (v') = f (r v + r' v') \in f (V_1)\), because \(f\) is linear.
By the proposition that for any vectors space, any nonempty subset of the vectors space is a vectors subspace if and only if the subset is closed under linear combination, \(f (V_1)\) is a vectors subspace of \(V_2\).
