# In Linear Algebra, what is a vector?

I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is.

I know in physics a vector is a geometric object that has a magnitude and a direction and it computer science a vector is a container that holds elements, expand, or shrink, but in linear algebra the definition of a vector isn’t too clear.

As a result, what is a vector in Linear Algebra?

In modern mathematics, there’s a tendency to define things in terms of what they do rather than in terms of what they are.

As an example, suppose that I claim that there are objects called “pizkwats” that obey the following laws:

• $$∀x.∀y.∃z.x+y=z\forall x. \forall y. \exists z. x + y = z$$
• $$∃x.x=0\exists x. x = 0$$
• $$∀x.x+0=0+x=x\forall x. x + 0 = 0 + x = x$$
• $$∀x.∀y.∀z.(x+y)+z=x+(y+z)\forall x. \forall y. \forall z. (x + y) + z = x + (y + z)$$
• $$∀x.x+x=0\forall x. x + x = 0$$

These rules specify what pizkwats do by saying what rules they obey, but they don’t say anything about what pizkwats are. We can find all sorts of things that we could call pizkwats. For example, we could imagine that pizkwats are the numbers 0 and 1, with addition being done modulo 2. They could also be bitstrings of length 137, with “addition” meaning “bitwise XOR.” Or they could be sets, with “addition” meaning “symmetric difference.” Each of these groups of objects obey the rules for what pizkwats do, but neither of them “are” pizkwats.

The advantage of this approach is that we can prove results about pizkwats knowing purely how they behave rather than what they fundamentally are. For example, as a fun exercise, see if you can use the above rules to prove that

$$∀x.∀y.x+y=y+x\forall x. \forall y. x + y = y + x$$.

This means that anything that “acts like a pizkwat” must support a commutative addition operator. Similarly, we could prove that

$$∀x.∀y.(x+y=0→x=y)\forall x. \forall y. (x + y = 0 \rightarrow x = y)$$.

The advantage of setting things up this way is that any time we find something that “looks like a pizkwat” in the sense that it obeys the rules given above, we’re guaranteed that it must have some other properties, namely, that it’s commutative and that every element has its own and unique inverse. We could develop a whole elaborate theory about how pizkwats behave and what pizkwats do purely based on the rules of how they work, and since we specifically never actually said what a pizkwat is, anything that we find that looks like a pizkwat instantly falls into our theory.

In your case, you’re asking about what a vector is. In a sense, there is no single thing called “a vector,” because a vector is just something that obeys a bunch of rules. But any time you find something that looks like a vector, you immediately get a bunch of interesting facts about it – you can ask questions about spans, about changing basis, etc. – regardless of whether that thing you’re looking at is a vector in the classical sense (a list of numbers, or an arrow pointing somewhere) or a vector in a more abstract sense (say, a function acting as a vector in a “vector space” made of functions.)

As a concluding remark, Grant Sanderson of 3blue1brown has an excellent video talking about what vectors are that explores this in more depth.