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# 1.1: What Are Vectors?

Here are some examples of things that can be added:

a)    Numbers: If $$x$$ and $$y$$ are numbers, then so is $$x+y$$.

b)    3-vectors: $$\begin{pmatrix}1\\ 1\\ 0\end{pmatrix} + \begin{pmatrix}0\\ 1\\ 1\end{pmatrix} =\begin{pmatrix}1\\ 2\\ 1\end{pmatrix}$$.

c)    Polynomials: If $$p(x)=1+x-2x^2+3x^3$$ and $$q(x)=x+3x^2-3x^3+x^4$$, then their sum $$p(x)+q(x)$$ is the new polynomial $$1+2x+x^2+x^4$$.

d)    Power series: If $$f(x)=1+x+\frac1{2!} x^2 + \frac1{3!} x^3 +\cdots$$ and $$g(x)=1-x+\frac1{2!} x^2 - \frac1{3!} x^3 +\cdots$$, then $$f(x)+g(x)=1+\frac1{2!} x^2 + \frac1{4!} x^4 +\cdots$$ is also a power series.

e)    Functions: If $$f(x)=e^x$$ and $$g(x)=e^{-x}$$, then their sum $$f(x)+g(x)$$ is the new function $$2\cosh x$$.

Stacks of numbers are not the only things that are vectors, as examples C, D, and E show. Because they "can be added'', you should now start thinking of all the above objects as vectors! In Chapter 5 we will give the precise rules that vector addition must obey.  In the above examples, however, notice that the vector addition rule stems from the rules for adding numbers.

When adding the same vector over and over, for example
$$x+x\, ,\: \ x+x+x\, ,\:\ x+x+x+x\, ,\, \ldots\, ,$$
we will write
$$2x\, ,\: \, 3x\, , \:\, 4x\, ,\, \ldots\, ,$$
respectively.  For example,
$$4\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}=\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}+\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}+\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}+\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}=\begin{pmatrix}4\\ 4\\ 0\end{pmatrix} .$$
Defining $$4x=x+x+x+x$$ is fine for integer multiples, but does not help us make sense of $$\tfrac{1}{3}x$$.  For the different types of vectors above, you can probably guess how to multiply a vector by a scalar, for example:
$$\tfrac{1}{3} \begin{pmatrix}1\\ 1\\ 0\end{pmatrix} = \begin{pmatrix}\tfrac{1}{3}\\ \tfrac{1}{3}\\ 0\end{pmatrix} .$$
In any given problem that you are planning to describe using vectors, you need to decide on a way to add and scalar multiply vectors.
In summary:

Vectors are things you can add and scalar multiply.