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Vector spaces
The familiar vectors in $\mathbb R^2$ and $\mathbb R^3$ can be added and multiplied by scalars. The same algebraic pattern appears in many other collections of objects. A vector space captures that shared structure.
A vector space $V$ over $\mathbb R$ is a set whose elements can be added and multiplied by real scalars, with the usual rules of linear arithmetic.
The required structure
There is a zero vector $\mathbf0$, every vector $\mathbf v$ has an additive opposite $-\mathbf v$, and operations satisfy rules such as
$$\mathbf u+\mathbf v=\mathbf v+\mathbf u,$$
$$a(\mathbf u+\mathbf v)=a\mathbf u+a\mathbf v,$$
$$(a+b)\mathbf v=a\mathbf v+b\mathbf v,$$
$$a(b\mathbf v)=(ab)\mathbf v.$$
These properties are what matter; the vectors themselves do not have to be arrows.
Vectors can be many kinds of objects
The set of polynomials of degree at most $2$ is a vector space. For example,
$$p(x)=1+2x-x^2$$
is a vector in that space, and polynomials can be added and scaled in the usual way.
Matrices of a fixed size also form a vector space, as do many spaces of functions.
Earlier ideas still apply
Once addition and scalar multiplication are available, linear combinations, span, linear independence, bases and dimension make sense exactly as they did for coordinate vectors.
Vector spaces therefore separate the essential linear structure from any particular representation.