719: For Linear Map from Finite-Dimensional Vectors Space, There Is Domain Subspace That Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Range by Restriction of Map

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description/proof of that for linear map from finite-dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to range by restriction of map

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of linear map.
• The reader knows a definition of %category name% isomorphism.
• The reader admits the proposition that any linear surjection from any finite-dimensional vectors space to any same-dimensional vectors space is a 'vectors spaces - linear morphisms' isomorphism.
• The reader admits the proposition that the image of any finite-dimensional vectors space under any linear map is a vectors space.

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Target Context

• The reader will have a description and a proof of the proposition that for any linear map from any finite-dimensional vectors space, there is a domain subspace that is 'vectors spaces - linear morphisms' isomorphic to the map range by the restriction of the map on the subspace domain.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V_1$: $\in \{\text{ the finite-dimensional }F\text{ vectors spaces }\}$
$V_2$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$f$: $V_1\to V_2$, $\in \{\text{ the linear maps }\}$
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Statements:
$\mathrm{\exists }V\in \{\text{ the subspaces of }{V}_1\}(f{|}_V:V\to f(V_1)\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})$
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2: Natural Language Description

For any field, $F$, any finite-dimensional $F$ vectors space, $V_1$, any $F$ vectors space, $V_2$, and any linear map, $f:V_1\to V_2$, there is a domain subspace, $V\subseteq V_1$, that is 'vectors spaces - linear morphisms' isomorphic to the map range, $f(V_1)$, by the restriction of $f$ on the $V$ domain.

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3: Proof

Whole Strategy: Step 1: see that $f(V_1)$ is a vectors space; Step 2: find a subset of any basis for $V_1$ whose space, $V$, surjectively maps to $f(V_1)$; Step 3: see that $V$ is same-dimensional with $f(V_1)$; Step 4: conclude the proposition.

Step 1:

By the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space, $f(V_1)$ is a vectors space.

Step 2:

In fact, the proof of the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space has found a subset of any basis for $V_1$, $\{b_1,b_2,...,b_r\}$, whose space, $V$, surjectively maps to $f(V_1)$.

Step 3:

In fact, the proof of the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space has shown that $V$ is same-dimensional with $f(V_1)$.

Step 4:

By the proposition that any linear surjection from any finite-dimensional vectors space to any same-dimensional vectors space is a 'vectors spaces - linear morphisms' isomorphism, $f{|}_V:V\to f(V_1)$ is a 'vectors spaces - linear morphisms' isomorphism.

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References

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