Item:Q2144 - Revision history

Rongzhou: Updated the item for vector space basis definition from Special:Update

2026-10-12T13:48:37Z

Updated the item for vector space basis definition from Special:Update

← Older revision		Revision as of 13:48, 12 October 2026
Property / instance of
	~~mathematical expression~~
Property / instance of: mathematical expression / rank
	~~Normal rank~~
Property / LaTeX source
	~~\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}~~
Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
	~~Normal rank~~
Property / note
	~~And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$~~
Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$ / rank
	~~Normal rank~~
Property / attributed to
	~~Rong Martin-Siebler ZHOU~~
Property / attributed to: Rong Martin-Siebler ZHOU / rank
	~~Normal rank~~
		Property / instance of
			mathematical expression
		Property / instance of: mathematical expression / rank
			Normal rank
		Property / LaTeX source
			\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}
		Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
			Normal rank
		Property / note
			And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$n$ is known as the '''dimension''' of $V$
		Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$n$ is known as the '''dimension''' of $V$ / rank
			Normal rank
		Property / attributed to
			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
			Normal rank

Rongzhou: Updated the item for vector space basis definition from Special:Update

2026-10-12T11:17:42Z

Updated the item for vector space basis definition from Special:Update

← Older revision		Revision as of 11:17, 12 October 2026
label / en		label / en
	vector space ~~base~~ definition		vector space basis definition
Property / instance of
	~~mathematical expression~~
Property / instance of: mathematical expression / rank
	~~Normal rank~~
Property / LaTeX source
	~~\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}~~
Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
	~~Normal rank~~
Property / note
	~~And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$~~
Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$ / rank
	~~Normal rank~~
Property / attributed to
	~~Rong Martin-Siebler ZHOU~~
Property / attributed to: Rong Martin-Siebler ZHOU / rank
	~~Normal rank~~
		Property / instance of
			mathematical expression
		Property / instance of: mathematical expression / rank
			Normal rank
		Property / LaTeX source
			\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}
		Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
			Normal rank
		Property / note
			And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$
		Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$ / rank
			Normal rank
		Property / attributed to
			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
			Normal rank

Rongzhou: Updated the item for vector space base definition from Special:Update

2026-10-12T07:32:43Z

Updated the item for vector space base definition from Special:Update

← Older revision		Revision as of 07:32, 12 October 2026
Property / instance of
	~~mathematical expression~~
Property / instance of: mathematical expression / rank
	~~Normal rank~~
Property / LaTeX source
	~~\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}~~
Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
	~~Normal rank~~
Property / note
	~~And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$~~
Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$ / rank
	~~Normal rank~~
Property / attributed to
	~~Rong Martin-Siebler ZHOU~~
Property / attributed to: Rong Martin-Siebler ZHOU / rank
	~~Normal rank~~
		Property / instance of
			mathematical expression
		Property / instance of: mathematical expression / rank
			Normal rank
		Property / LaTeX source
			\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}
		Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
			Normal rank
		Property / note
			And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$
		Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$\n\n$\\text{len}(B)$ is known as the '''dimension''' of $V$ / rank
			Normal rank
		Property / attributed to
			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
			Normal rank

Rongzhou: Updated the item for vector space base definition from Special:Update

2026-10-12T07:28:34Z

Updated the item for vector space base definition from Special:Update

← Older revision		Revision as of 07:28, 12 October 2026
Property / instance of
	~~mathematical expression~~
Property / instance of: mathematical expression / rank
	~~Normal rank~~
Property / LaTeX source
	~~\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}~~
Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
	~~Normal rank~~
Property / note
	~~And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$~~
Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$ / rank
	~~Normal rank~~
Property / attributed to
	~~Rong Martin-Siebler ZHOU~~
Property / attributed to: Rong Martin-Siebler ZHOU / rank
	~~Normal rank~~
		Property / instance of
			mathematical expression
		Property / instance of: mathematical expression / rank
			Normal rank
		Property / LaTeX source
			\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}
		Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
			Normal rank
		Property / note
			And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$
		Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\n\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$ / rank
			Normal rank
		Property / attributed to
			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
			Normal rank

Rongzhou: Added content item: vector space base definition

2026-10-12T07:27:33Z

Added content item: vector space base definition

← Older revision		Revision as of 07:27, 12 October 2026
Property / instance of
	~~mathematical expression~~
Property / instance of: mathematical expression / rank
	~~Normal rank~~
Property / LaTeX source
	~~\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}~~
Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
	~~Normal rank~~
Property / note
	~~And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$~~
Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$ / rank
	~~Normal rank~~
Property / attributed to
	~~Rong Martin-Siebler ZHOU~~
Property / attributed to: Rong Martin-Siebler ZHOU / rank
	~~Normal rank~~
		Property / instance of
			mathematical expression
		Property / instance of: mathematical expression / rank
			Normal rank
		Property / LaTeX source
			\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}
		Property / LaTeX source: \\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle\|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\} / rank
			Normal rank
		Property / note
			And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$
		Property / note: And $\\vec u_1,\\dots,\\vec u_n$ are [[linearly independent]],\nThen $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a base of $V$ / rank
			Normal rank
		Property / attributed to
			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
			Normal rank

Rongzhou: Added content item: vector space base definition

2026-10-12T07:27:33Z

Added content item: vector space base definition

Show changes