While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commutativity of addition, etc.). Is there any widely used vector space in which alternative functions are used as addition/multiplication?
Linear Algebra – Alternative Definitions of Vector Addition and Scalar Multiplication
linear algebravector-spaces
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Yes. But you don't need to add closure in these definitions For groups, for example, notice that an operation is, first of all, a function $\cdot :G \times G \to G$. And that its codomain is $G$ itself.
A vector space is a $4-$tuple $(\mathcal{V},{\Bbb K}, +, \cdot)$, where $$+: \mathcal{V}^2 \to \mathcal{V} \quad \text{and} \quad\cdot:\mathbb{K} \times \mathcal{V} \to \mathcal{V}$$ are the operations. The structure of a vector space is much richer than that of a group. A vector space has two operations and a underlying a field, while a group is only the set with one operation (satisfying conditions you well know). Given a vector space $(\mathcal{V},{\Bbb K}, +, \cdot)$, $(\mathcal{V},+)$ is an abelian group, always. Answering 4. along, given a field $\Bbb K$, $\Bbb K^n$ is both a vector field and an additive group, with respect to the operations of $\Bbb K$.
Vectors are elements of a vector space. It is just a name. Examples of vector spaces are:
- Polynomials with degree less or equal to $n$, with real coefficients: $\mathcal{P}_n(\Bbb R)$.
- All continuous functions from $[0,1]$ to $\Bbb R$: $\mathscr{C}^0([0,1],\Bbb R)$
- $\Bbb R^n$ itself.
- Matrices with real coefficients: $\mathbb{M}_{n \times m}(\Bbb R)$.
and a lot more stuff. I used $\Bbb R$ for concreteness, in general you can take an arbitrary field (for polynomials, matrices, etc). So a vector can be an arrow, a function, a polynomial, a matrix...
Elements of a real vectorspace certainly have direction, but they don't really have a magnitude. Well actually, they... kind-of have a magnitude. But for a proper magnitude, you need further structure, such as a norm or inner product. Let me explain.
Vector Spaces.
Suppose $V$ is a real vectorspace.
Definition 0. Given a vectors $x,y \in V$, we say that $x$ and $y$ have the same direction iff:
- there exists $r \in \mathbb{R}_{\geq 0}$ such that $x = ry,$ and
- there exists $r \in \mathbb{R}_{\geq 0}$ such that $y = rx$.
(The $r$'s don't have to be the same.)
This induces an equivalence relation on $V$, so we get a partitioning of $V$ into cells. Each cell is an open ray, so long as we regard $\{0\}$ as an open ray. You may wish to exclude $\{0\}$ from its privileged position as a ray, in which case you should only deal with non-zero vectors; that is, you need to be dealing with $V \setminus \{0\}$ rather than $V$.
Irrespective of which conventions are used, we can make sense of direction using these ideas:
Definition 1. The direction of $x \in V$ is the unique open ray $R \subseteq V$ such that $x \in R$.
Notice that the equivalence relation of having the same direction is preserved under scalar multiplication; what I mean is that if $v$ and $w$ have the same direction, then $av$ and $aw$ have the same direction, for any $a \in \mathbb{R}$. Geometrically, this means that if we scale a ray, we'll end up with a subset of another ray.
As for magnitude; well, if you choose a ray $R \subseteq V$, then we can partially order $R$ as follows. Given $x,y \in R$, we define that $x \geq y$ iff $x = ry$ for some $r \in \mathbb{R}_{\geq 1}$. So some vectors along this ray are longer than others, hence magnitude.
Inner Product Spaces.
Actually, this isn't the whole story. The problem with vector spaces is that if $x$ and $y$ don't belong to the same ray (nor to the the "negatives" of each others rays), then there's no way of comparing the magnitudes of $x$ and $y$. We can't say which is longer! Now there are mathematical situations where this limitation is desirable, but physically, you probably don't want this. A related issue is that you can't really make sense of angles in a (mere) vector space; at least, not without some further structure.
For this reason, when physicists say "vector", what they usually mean is "element of a finite-dimensional inner-product space." This is a (finite-dimensional) vector space $V$ with further structure; in particular, it comes equipped with a function
$$\langle-,-\rangle : V \times V \rightarrow \mathbb{R}$$
that is required to satisfy certain axioms resembling the dot product. Especially important for us is that these axioms include a "non-negativity" condition:
$$\langle x,x\rangle \geq 0$$
Using this, we can define the magnitude of vectors as follows.
Definition 2. Suppose $V$ is a real inner product space. Then the norm (or "magnitude") of $x \in V$, denoted $\|x\|$, is defined a follows:
$$\|x\| = \langle x,x\rangle^{1/2}$$
This allows us to compare the magnitudes of vectors that don't live in the same ray; we simply define that $x \geq y$ means $\|x\| \geq \|y\|.$ When confined to a single ray, this agrees with our earlier definition! Be careful though, because the relation $\geq$ we just defined is only a preorder.
In fact, the inner product gives us more than just magnitudes; it also gives angles!
Definition 3. Suppose $V$ is a real inner product space. Then the angle between of $x,y \in V$, denoted $\mathrm{ang}(x,y)$, is defined a follows:
$$\mathrm{ang}(x,y) = \cos^{-1}\left(\frac{\langle x,y\rangle}{\|x\|\|y\|}\right)$$
It can be shown that vectors $x$ and $y$ have the same direction (in the sense described at the beginning of my post) iff the angle between them is $0$. In fact, you can modify the above definition so that it defines the angle between any two non-zero open rays. In this case, it turns out that two rays are equal iff the angle between them is $0$.
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My favorite example is the set of subsets of a set under the operation of symmetric difference (otherwise known as bitwise XOR). This forms a vector space over the finite field $\mathbb{F}_2$. This example is important in computer science, coding theory, combinatorics, ...