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# 4.1: Addition and Scalar Multiplication in $$\mathbb{R}^{n}$$

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A simple but important property of $$n$$-vectors is that we can $$\textit {add}$$ $$n$$-vectors and $$\textit {multiply}$$ $$n$$-vectors by a scalar:

Definition

Given two $$n$$-vectors $$a$$ and $$b$$ whose components are given by

$$a=\begin{pmatrix}a^{1} \\ \vdots \\ a^{n}\end{pmatrix} ~and ~b=\begin{pmatrix}b^{1} \\ \vdots \\ b^{n}\end{pmatrix}$$ their $$\textit{sum}$$ is

$$a+b := \begin{pmatrix} a^1+b^1 \\ \vdots \\ a^n+b^n\end{pmatrix}\, .$$

Given a scalar $$\lambda$$, the $$\textit{scalar multiple}$$

$$\lambda a := \begin{pmatrix}\lambda a^{1} \\ \vdots \\ \lambda a^{n}\end{pmatrix}\, .$$

Example 43

Let

$$a=\begin{pmatrix}1\\2\\3\\4\end{pmatrix} ~and ~b = \begin{pmatrix}4\\3\\2\\1\end{pmatrix}\, .$$

Then, for example,

$$a+b= \begin{pmatrix}5\\5\\5\\5\end{pmatrix} and 3a - 2b= \begin{pmatrix}-5\\0\\5\\10\end{pmatrix}\, .$$

A special vector is the $$\textit{zero vector}$$. All of its components are zero:

$$0=\begin{pmatrix}0\\ \vdots \\ 0\end{pmatrix}\, .$$

In Euclidean geometry---the study of $$\mathbb{R}^{n}$$ with lengths and angles defined as in section 4.3---$$n$$-vectors are used to label points $$P$$ and the zero vector labels the origin $$O$$. In this sense, the zero vector is the only one with zero magnitude, and the only one which points in no particular direction.