Definition: vector space hyperplane (Q2146): Difference between revisions
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Added content item: Definition: vector space hyperplane |
Added content item: Definition: vector space hyperplane |
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| Property / instance of | |||
| Property / instance of: mathematical expression / rank | |||
| Property / LaTeX source | |||
| Property / LaTeX source: P=\\left\\{ \\vec p + \\sum_{i=1}^{k} \\lambda_{i}\\vec v_{i}\\, |\\, \\vec p,\\vec v_i \\in V,k < \\dim(V), \\lambda_{i} \\in \\mathbb{F} \\right\\} \\\\ / rank | |||
| Property / note | |||
| Property / note: Then:\n\nP is a hyperplane of dimension $k$ in $V \\iff \\{\\vec v_i\\}$ is [[linearly independent#vectors|linearly independent]] / 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 | |||
P=\\left\\{ \\vec p + \\sum_{i=1}^{k} \\lambda_{i}\\vec v_{i}\\, |\\, \\vec p,\\vec v_i \\in V,k < \\dim(V), \\lambda_{i} \\in \\mathbb{F} \\right\\} \\\\ | |||
| Property / LaTeX source: P=\\left\\{ \\vec p + \\sum_{i=1}^{k} \\lambda_{i}\\vec v_{i}\\, |\\, \\vec p,\\vec v_i \\in V,k < \\dim(V), \\lambda_{i} \\in \\mathbb{F} \\right\\} \\\\ / rank | |||
Normal rank | |||
| Property / note | |||
Then:\n\nP is a hyperplane of dimension $k$ in $V \\iff \\{\\vec v_i\\}$ is [[linearly independent#vectors|linearly independent]] | |||
| Property / note: Then:\n\nP is a hyperplane of dimension $k$ in $V \\iff \\{\\vec v_i\\}$ is [[linearly independent#vectors|linearly independent]] / rank | |||
Normal rank | |||
| Property / attributed to | |||
| Property / attributed to: Rong Martin-Siebler ZHOU / rank | |||
Normal rank | |||
Revision as of 09:20, 12 October 2026
No description defined
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Definition: vector space hyperplane |
No description defined |
Statements
P=\\left\\{ \\vec p + \\sum_{i=1}^{k} \\lambda_{i}\\vec v_{i}\\, |\\, \\vec p,\\vec v_i \\in V,k < \\dim(V), \\lambda_{i} \\in \\mathbb{F} \\right\\} \\\\
0 references
Then:\n\nP is a hyperplane of dimension $k$ in $V \\iff \\{\\vec v_i\\}$ is [[linearly independent#vectors|linearly independent]]
0 references