Vector subspace - Revision history

Rongzhou at 11:03, 12 October 2026

2026-10-12T11:03:02Z

← Older revision		Revision as of 11:03, 12 October 2026
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	Let $V$ be a vector space over $\mathbb{F}$.		Let $V$ be a vector space over $\mathbb{F}$.

	~~Set $U$ is a vector subspace of $V$ generated by $U$ if~~:		If:

	{{#content:Q2002}}		{{#content:Q2002}}

	~~Set $\{u_1,\dots,u_n\}$ is then known as a '''generating set''' of $U$~~

	== Intersection and union ==		== Intersection and union ==

	If $F,G$ ~~are vector subspaces of vector space~~ $V$:		For vector subspaces $F,G$ $\in V$

	$$		$$
	\vec u \in F \~~land~~ G \iff \vec u \in F \land \vec u \in G \text{ intersection}\\		\vec u \in F \cap G \iff \vec u \in F \land \vec u \in G \text{ intersection}\\
	\vec u \in F \~~lor~~ G \iff \vec u \in F \lor \vec u \in G \text{ union}		\vec u \in F \cup G \iff \vec u \in F \lor \vec u \in G \text{ union}
	$$		$$

	== Sum ==		== Sum ==

	If $F,G$ ~~are vector subspaces of vector space~~ $V$:		For vector subspaces $F,G$ $\in V$

	$$		$$
	F+G = \{\vec u + \vec v \|\vec u \in F, \vec v \in G \}		F+G = \{\vec u + \vec v \|\vec u \in F, \vec v \in G \}
			$$

			=== Property ===

			$$
			\dim(F+G)=\dim(F)+\dim(G)-\dim(F \cap G)
	$$		$$

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	$$		$$

	for $U=\{u_1,\cdots, u_n\} \in~~$ any vector space $~~V$.		for $U=\{u_1,\cdots, u_n\} \in V$.

	== Supplementary subspaces ==		== Supplementary subspaces ==

	For vector subspaces $F,G$ ~~of vector space~~ $V$, if		For vector subspaces $F,G$ $\in V$, if

	* $F+G=E$, and		* $F+G=E$, and
	* $F \~~land~~ G = \{ \vec 0_V \}$		* $F \cap G = \{ \vec 0_V \}$

	Then $F,G$ are supplementary vector subspaces of $V$, ~~alternatively expressed~~ as $E=F \oplus G$		Then $F,G$ are '''supplementary vector subspaces''' of $V$, noted as $E=F \oplus G$

Rongzhou at 07:10, 12 October 2026

2026-10-12T07:10:26Z

← Older revision		Revision as of 07:10, 12 October 2026
Line 1:		Line 1:
	Let $V$ be a vector space.		Let $V$ be a vector space over $\mathbb{F}$.

	Set $U$ is a vector subspace of $V$ generated by $U$ if:		Set $U$ is a vector subspace of $V$ generated by $U$ if:

Rongzhou at 08:01, 5 October 2026

2026-10-05T08:01:17Z

← Older revision		Revision as of 08:01, 5 October 2026
Line 5:		Line 5:
	{{#content:Q2002}}		{{#content:Q2002}}

	Set ${u_1,\dots,u_n\}$ is then known as a '''generating set''' of $U$		Set $\{u_1,\dots,u_n\}$ is then known as a '''generating set''' of $U$

	== Intersection and union ==		== Intersection and union ==
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	$$		$$
	\vec u \in F \land G \iff \vec u in F \land \vec u \in G \text{ intersection}\\		\vec u \in F \land G \iff \vec u \in F \land \vec u \in G \text{ intersection}\\
	\vec u \in F \lor G \iff \vec u in F \lor \vec u \in G \text{ union}		\vec u \in F \lor G \iff \vec u \in F \lor \vec u \in G \text{ union}
	$$		$$

Rongzhou at 07:59, 5 October 2026

2026-10-05T07:59:58Z

← Older revision		Revision as of 07:59, 5 October 2026
Line 4:		Line 4:

	{{#content:Q2002}}		{{#content:Q2002}}

			Set ${u_1,\dots,u_n\}$ is then known as a '''generating set''' of $U$

			== Intersection and union ==

			If $F,G$ are vector subspaces of vector space $V$:

			$$
			\vec u \in F \land G \iff \vec u in F \land \vec u \in G \text{ intersection}\\
			\vec u \in F \lor G \iff \vec u in F \lor \vec u \in G \text{ union}
			$$

			== Sum ==

			If $F,G$ are vector subspaces of vector space $V$:

			$$
			F+G = \{\vec u + \vec v \|\vec u \in F, \vec v \in G \}
			$$

			== Composition ==

			$$
			\operatorname{span}\{u_1,\dots,u_n\} = \operatorname{span}\{u_1\} + \cdots + \operatorname{span}\{u_n\}
			$$

			for $U=\{u_1,\cdots, u_n\} \in$ any vector space $V$.

			== Supplementary subspaces ==

			For vector subspaces $F,G$ of vector space $V$, if

			* $F+G=E$, and
			* $F \land G = \{ \vec 0_V \}$

			Then $F,G$ are supplementary vector subspaces of $V$, alternatively expressed as $E=F \oplus G$

Rongzhou at 07:40, 5 October 2026

2026-10-05T07:40:07Z

← Older revision	Revision as of 07:40, 5 October 2026
Line 1:	Line 1:
		Let $V$ be a vector space.

		Set $U$ is a vector subspace of $V$ generated by $U$ if:

		{{#content:Q2002}}

Rongzhou: Created the wiki page for Vector subspace

2026-10-05T07:26:42Z

Created the wiki page for Vector subspace

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