736: Projection of Vector into Vectors Subspace w.r.t. Complementary Subspace

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definition of projection of vector into vectors subspace w.r.t. complementary subspace

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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: Note

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Starting Context

• The reader knows a definition of complementary subspace of vectors subspace.

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Target Context

• The reader will have a definition of projection of vector into vectors subspace with respect to complementary subspace

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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:
$F$: $\in \{\text{ the fields }\}$
$V^{\prime }$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$V$: $\in \{\text{ the vectors subspaces of }{V}^{\prime }\}$
$\tilde{V}$: $\in \{\text{ the complementary subspaces of }V\}$
$v^{\prime }$: $=v+\tilde{v}$, $\in V^{\prime }$
$\ast v$:
//

Conditions:
$v\in V\wedge \tilde{v}\in \tilde{V}$
//

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2: Natural Language Description

For any field, $F$, any $F$ vectors space, $V^{\prime }$, any vectors subspace of $V^{\prime }$, $V$, any complementary subspace of $V$, $\tilde{V}$, and any element, $v^{\prime }=v+\tilde{v}\in V^{\prime }$ such that $v\in V$ and $\tilde{v}\in \tilde{V}$, $v$

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3: Note

Once $V$ and $\tilde{V}$ are specified, the projection is uniquely determined, as is described in Note for the definition of complementary subspace of vectors subspace.

A complementary subspace is need to be specified in order to talk about projection, because the projection depends on the choice of complementary subspace.

For example, when $V^{\prime }$ is the span of the basis, $(e_1,e_2)$, and $V$ is the span of the basis, $(e_1)$, if $\tilde{V}$ as the span of the basis, $(e_2)$, is the complementary subspace, the projection of $e_1+e_2$ is $e_1$, but if $\tilde{V}$ as the span of the basis, $(e_1+e_2)$, is the complementary subspace, the projection of $e_1+e_2$ is $0$.

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References

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