4: Vectors in Space, n-Vectors
- Page ID
- 1726
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:
\[\begin{pmatrix}7 \\4 \\ 2\\ 5 \end{pmatrix} \neq \begin{pmatrix}7 \\2 \\4 \\5 \end{pmatrix} .\]
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 \(r_3\). (CC BY-SA 3.0; Claudio Rocchini)
Contributor
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)