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Mathematics LibreTexts

4: Vectors in Space, n-Vectors


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To continue our linear algebra journey, we must discuss \(n\)-vectors with an arbitrarily large number of components. The simplest way to think about these is as  ordered  lists of numbers, 

\[a=\begin{pmatrix}a^{1} \\ \vdots \\ a^{n}\end{pmatrix} .\]

\(\textit {Do not be confused by our use of a superscript to label components of a vector. Here \(a^2\) denotes the second component of the vector \(a\), rather than the number $a$ squared!}\) We emphasize that order matters: 

Example 42: Order of Components Matters

\[\begin{pmatrix}7 \\4 \\ 2\\ 5 \end{pmatrix} \neq \begin{pmatrix}7 \\2 \\4 \\5 \end{pmatrix} .\]

The set of all \(n\)-vectors is denoted \(\mathbb{R}^n\). As an equation

\[ {\mathbb{R}}^n :=\left\{ \begin{pmatrix}a^1 \\ \vdots\ \  \\ a^n\end{pmatrix} \middle\vert \,  a^1,\dots, a^n \in \mathbb{R} \right\} \,.\]

Thumbnail: The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors \(r_1\), \(r_2\), and \(r_3\). Image used with permission (CC BY-SA 3.0; Claudio Rocchini)