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

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A description/proof of that for linear map from finite dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to image 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: Description
• 2: Proof

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Starting Context

• The reader knows a definition of 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 linear map from any finite dimensional vectors space 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 image 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: Description

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

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2: Proof

By the the proposition that the image of any linear map from any finite dimensional vectors space is a vectors space, $f(V_1)$ is a vectors space. As is shown in the proof of the proposition, there is a subset of any basis for $V_1$, $b_1,b_2,...,b_r$, whose space surjectively maps to $f(V_1)$. As the domain subspace is of the same dimension with $f(V_1)$, 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, the linear surjection from the domain subspace to $f(V_1)$ (the restriction of $f$ to the domain subspace) is a 'vectors spaces - linear morphisms' isomorphism.

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References

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