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# 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:

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

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)