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For the last couple years I have been taught that any thing with a direction and a magnitude is considered a vector i.e. $\vec V$

But recently a friend went up and told me that this intuition is completely wrong. Anything that satisfies the axioms of vector space is a vector. This extends to matrices, polynomials, and functions.

I have always treated functions and vectors as completely different things...because functions do not have magnitude and direction.

Can someone help me reconcile the fact that functions do not have magnitude and direction but are still considered a vector?

Thanks

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    $\begingroup$ On a side note, the type of vector spaces in which it makes sense to talk about magnitude and direction are those vector spaces which are induced by an inner product. (If they are also complete, we call them Hilbert spaces). It turns out all finite dimensional vector spaces are equivalent to standard euclidean spaces. However, there are some infinite dimensional ones where we can meaningfully define "angles" via an inner product, even though they don't look like 'arrows' per se. $\endgroup$ – Alan Feb 14 '15 at 7:52
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    $\begingroup$ You were looking at vectors from the physical point of view. Mathematically, an element is called a vector whenever the set it belongs to is a vector space! For example the set of sequences associated with the operation + and the external operation . (usual multiplication) can be considered as a vector space, hence every sequence belonging to this set is considered a vector! $\endgroup$ – Mistos Feb 14 '15 at 10:08
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A vector is an element of a vector space. If a vector space has an inner product that inner product gives each vector $x$ a magnitude ($\Vert x\Vert=\sqrt{\langle x,x\rangle}$) and each pair of vectors $x$, $y$ an angle $\arccos(\langle x,y\rangle/\Vert x\Vert\Vert y\Vert)$.

Mathematicians consider inner products optional for vector spaces, although many examples of vector spaces have inner products.

  • Let $C([0,1])$ be the space of continuous functions $[0,1]\to\mathbb C$. The following equation defines an inner product on this space. $$ \langle f,g\rangle=\int\limits_0^1 f(x)\overline{g(x)}dx$$

  • The following formula defines en inner products for polynomials $p(x)=p_0+p_1x+p_2x^2+\dotsm+p_mx^m$ and $q(x)=q_0+q_1x+q_2x^2+\dotsm+q_nx^n$ with coefficients in $\mathbb C$. $$ \langle p,q\rangle = \sum^{\min(m,n)}_{i=0} p_i\overline{q_i}$$

  • Matrices have the Frobenius inner product of which the dot products on $\mathbb C^n$ is a special case. $$ \langle M,N\rangle =\mathrm{trace}(MN^*)=\sum_{i,j} M^i_j\overline{N^i_j}$$

The examples help to show that inner products are ambiguous, which is the reason that they are optional. The inner product on $C([0,1])$ diverges on some pairs of functions in $C(]0,1[)$ despite the (superficial) similarity between $[0,1]$ and $]0,1[$. Each vector space (over $\mathbb R$ or $\mathbb C$) has many different inner products. Polynomials have the following alternative for example. $$ \langle p,q\rangle = \sum^{\min(m,n)}_{i=0} \frac1{i!}p_i\overline{q_i}$$

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In basic calculus, to make it easier to the students, a "vector" is simply an arrow. It may be better to call them "arrows", to avoid this confusion. But anyway, the terminology comes from physics.

But in mathematics, there is something called a "vector space", and those things that belong to the vector space are called "vectors". It can be very abstract things, not just functions or matrices. It is possible to think of arrows as a type of vector space, but there are many more.

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The notion of magnitude and direction is used when vectors are explained in terms of physics, or more generally elements of $\mathbb{R}^3$. In mathematics a vector space is a collection of elements that is closed under addition of its elements, and multiplication of its elements by scalars from a field (say, $\mathbb{R}$, or $\mathbb{C}$).

There are many, many, and many examples of vector spaces since the structure is quite a flexible one but whether you describe a function as a vector or simply an arrow as a vector really depends on the space in which you're viewing and how the object relates to all the other objects in your space.

The short answer to your question is that it's perfectly reasonable to call arrows in 3-dimensional space vectors, but in the context of vector spaces there are many different types of vectors and they all depend on the space in which you're viewing them.

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