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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

( \newcommand{\kernel}{\mathrm{null}\,}\)

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,

$\begin{matrix} (4.1) & a = (\begin{matrix} a^{1} \\ ⋮ \\ a^{n} \end{matrix}) . \end{matrix}$

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

$\begin{matrix} (4.2) & (\begin{matrix} 7 \\ 4 \\ 2 \\ 5 \end{matrix}) \neq (\begin{matrix} 7 \\ 2 \\ 4 \\ 5 \end{matrix}) . \end{matrix}$

The set of all $n$ -vectors is denoted $R^{n}$ . As an equation

$\begin{matrix} (4.3) & R^{n} := {(\begin{matrix} a^{1} \\ ⋮ \\ a^{n} \end{matrix}) | a^{1}, \dots, a^{n} \in R} . \end{matrix}$

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- $R^{S}$
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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