vector space basis definition (Q2144): Difference between revisions
Jump to navigation
Jump to search
Updated the item for vector space base definition from Special:Update |
Updated the item for vector space basis definition from Special:Update |
||
| label / en | label / en | ||
vector space | vector space basis definition | ||
| Property / instance of | |||
| Property / instance of: mathematical expression / rank | |||
| Property / LaTeX source | |||
| 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 | |||
| Property / note | |||
| 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 | |||
| Property / attributed to | |||
| Property / attributed to: Rong Martin-Siebler ZHOU / rank | |||
| Property / instance of | |||
| 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 | |||
| Property / attributed to: Rong Martin-Siebler ZHOU / rank | |||
Normal rank | |||
Revision as of 11:17, 12 October 2026
No description defined
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | vector space basis definition |
No description defined |
Statements
\\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\\}
0 references
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 $V$
0 references