737: Projection from Vectors Space into Subspace w.r.t. Complementary Subspace Is Linear Map and Image of Any Subspace Is Subspace|T.B.P.

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description/proof of that projection from vectors space into subspace w.r.t. complementary subspace is linear map and image of any subspace is subspace

Topics

About: vectors space

Starting Context

The reader knows a definition of projection of vector into vectors subspace with respect to complementary subspace.

Target Context

The reader will have a description and a proof of the proposition that any projection from any vectors space into any subspace with respect to any complementary subspace is a linear map, and the image of any subspace under the projection is a subspace.

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 {the fields}

V^{'}

\in {the F vectors spaces}

V

\in {the vectors subspaces of V^{'}}

\tilde{V}

\in {the complementary subspaces of V}

f

: V^{'} \to V

= the projection map into V with respect to \tilde{V}

V^{″}

\in {the vectors subspaces of V^{'}}

//

Statements:

f \in {the linear maps}

\land

f (V^{″}) \in {the subspaces of V^{'}}

2: Natural Language Description

For any field,

F

, any

F

vectors space,

V^{'}

, any vectors subspace of

V^{'}

V

, any complementary subspace of

V

\tilde{V}

, the projection map into

V

with respect to

\tilde{V}

f : V^{'} \to V

, and any vectors subspace,

V^{″} \subseteq V^{'}

f

is a linear map, and

f (V^{″})

is a vectors subspace of

V^{'}

3: Proof

Whole Strategy: Step 1: choose any

v_{1}^{'}, v_{2}^{'} \in V^{'}

and any

r_{1}, r_{2} \in F

, and see that

f (r_{1} v_{1}^{'} + r_{2} v_{2}^{'}) = r_{1} f (v_{1}^{'}) + r_{2} f (v_{2}^{'})

; Step 2: choose any

v_{1}, v_{2} \in f (V^{″})

, and see that

r_{1} v_{1} + r_{2} v_{2} \in f (V^{″})

.

Step 1:

For any

v_{1}^{'}, v_{2}^{'} \in V^{'}

and any

r_{1}, r_{2} \in F

f (r_{1} v_{1}^{'} + r_{2} v_{2}^{'}) = r_{1} f (v_{1}^{'}) + r_{2} f (v_{2}^{'})

v_{j}^{'} = v_{j} + {\tilde{v}}_{j}

where

v_{j} \in V

and

{\tilde{v}}_{j} \in \tilde{V}

r_{1} v_{1}^{'} + r_{2} v_{2}^{'} = r_{1} (v_{1} + {\tilde{v}}_{1}) + r_{2} (v_{2} + {\tilde{v}}_{2}) = r_{1} v_{1} + r_{2} v_{2} + r_{1} {\tilde{v}}_{1} + r_{2} {\tilde{v}}_{2}

f (r_{1} v_{1}^{'} + r_{2} v_{2}^{'}) = r_{1} v_{1} + r_{2} v_{2} = r_{1} f (v_{1}^{'}) + r_{2} f (v_{2}^{'})

.

So, yes.

Step 2:

For any

v_{1}, v_{2} \in f (V^{″})

r_{1} v_{1} + r_{2} v_{2} \in f (V^{″})

?

There is a

v_{j}^{″} \in V^{″}

such that

v_{j} = f (v_{j}^{″})

r_{1} v_{1}^{″} + r_{2} v_{2}^{″} \in V^{″}

f (r_{1} v_{1}^{″} + r_{2} v_{2}^{″}) = r_{1} f (v_{1}^{″}) + r_{2} f (v_{2}^{″}) = r_{1} v_{1} + r_{2} v_{2} \in f (V^{″})

.

So, yes.

References

The Bias Planet
Definitions and Propositions
School Mathematics from Higher Viewpoints
To Disentangle Confusing Terms or Discourses
Information Tables
Exploiting LibreOffice or OpenOffice
Exposing the Java Programming Language
Exposing C++
Exposing Python
Exposing C#
Exposing Gradle
Exposing Git
UNO Dispatch Commands

2024-08-18

737: Projection from Vectors Space into Subspace w.r.t. Complementary Subspace Is Linear Map and Image of Any Subspace Is Subspace