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4: Vectors in Space, n-Vectors

Last updated

May 28, 2023
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- 3.5: Review Problems
- 4.1: Addition and Scalar Multiplication in Rⁿ

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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

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

4.1: Addition and Scalar Multiplication in Rⁿ
A simple but important property of n-vectors is that we can add n-vectors and multiply n-vectors by a scalar.
4.2: Hyperplanes
4.3: Directions and Magnitudes
4.4: Vectors, Lists and Functions- $\mathbb{R}^{S}$
4.5: Review Problems

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)

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David Cherney, Tom Denton, and Andrew Waldron (UC Davis)

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