(Untitled) | (Untitled) | (Untitled) | (Untitled) | (Untitled) | Examples |

IEVref: | 102-03-37 | ID: | |

Language: | en | Status: Standard | |

Term: | determinant, <of <i>n</i> vectors> | ||

Synonym1: | |||

Synonym2: | |||

Synonym3: | |||

Symbol: | |||

Definition: | for an ordered set of n vectors in an n-dimensional space with a given base, scalar attributed to this set by the unique multilinear form taking the value 0 when the vectors are linearly dependent and the value 1 for the base vectorsNote 1 to entry: When the coordinates of the $\mathrm{det}\text{\hspace{0.17em}}({U}_{1}\text{,}{U}_{2}\text{,}\dots \text{,}{U}_{n}\text{)}=\left|\begin{array}{cccc}{U}_{11}& {U}_{12}& \cdots & {U}_{1n}\\ {U}_{21}& {U}_{22}& \cdots & {U}_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ {U}_{n1}& {U}_{n2}& \cdots & {U}_{nn}\end{array}\right|$ Note 2 to entry: According to the sign of the determinant, the set of vectors and the given base have the same orientation or opposite orientations. Note 3 to entry: For the three-dimensional Euclidean space, the determinant of three vectors is the scalar triple product of the vectors. | ||

Publication date: | 2008-08 | ||

Source: | |||

Replaces: | |||

Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

CO remarks: | |||

TC/SC remarks: | |||

VT remarks: | |||

Domain1: | |||

Domain2: | |||

Domain3: | |||

Domain4: | |||

Domain5: |

Note 1 to entry: When the coordinates of the *n* vectors ${U}_{1}\text{,}{U}_{2}\text{,}\dots ,\text{}{U}_{n}$ are arranged as columns or rows of an $n\times n$ matrix, the determinant of the vectors is equal to the determinant of the matrix:

$\mathrm{det}\text{\hspace{0.17em}}({U}_{1}\text{,}{U}_{2}\text{,}\dots \text{,}{U}_{n}\text{)}=\left|\begin{array}{cccc}{U}_{11}& {U}_{12}& \cdots & {U}_{1n}\\ {U}_{21}& {U}_{22}& \cdots & {U}_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ {U}_{n1}& {U}_{n2}& \cdots & {U}_{nn}\end{array}\right|$

Note 2 to entry: According to the sign of the determinant, the set of vectors and the given base have the same orientation or opposite orientations.

Note 3 to entry: For the three-dimensional Euclidean space, the determinant of three vectors is the scalar triple product of the vectors.