1091: Projection from Vectors Space into Vectors Subspace

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definition of projection from vectors space into vectors 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: Note

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Starting Context

• The reader knows a definition of linear map.
• The reader knows a definition of %field name% vectors space.

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Target Context

• The reader will have a definition of projection from vectors space into vectors 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:
$F$: $\in \{\text{ the fields }\}$
$V^{\prime }$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$V$: $\in \{\text{ the vectors subspaces of }{V}^{\prime }\}$
$\ast f$: $:V^{\prime }\to V$, $\in \{\text{ the linear maps }\}$
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Conditions:
$\mathrm{\forall }v\in V(f(v)=v)$
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2: Note

Inevitably, $f\circ f=f$, because for each $v^{\prime }\in V^{\prime }$, $f(v^{\prime })\in V$, so, $f\circ f(v^{\prime })=f(v^{\prime })$.

This definition does not require $V^{\prime }$ equipped with any inner product.

$V^{\prime }$ or $V$ does not need to be finite-dimensional.

Any vectors subspace, $W\subseteq V^{\prime }$, such that for each $w\in W$, $f(w)=0$, is said to be "perpendicular to $V$ with respect to $f$": that certainly depends on the choice of $f$.

$W$'s being perpendicular to $V$ with respect to $f$ of course does not imply $V$'s being perpendicular to $W$, because without any projection into $W$ not specified, it does not make any sense.

When $V^{\prime }$ has an inner product, the projection into each vectors subspace is often (rather implicitly) defined by the definition of orthogonal projection from vectors space with inner product into vectors subspace, and whether $V$ is perpendicular to $W$ is often talked about without explicitly mentioning the projection.

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References

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