At its heart, a vector space is a collection of objects called vectors, bundled with a set of rules for how they interact. You're likely familiar with vectors as arrows with a specific length and direction, but in linear algebra, the idea is much broader. Vectors can be numbers, functions, or even polynomials. What matters are the rules.

For a collection of objects to qualify as a vector space, two basic operations must be defined: vector addition and scalar multiplication. You need to be able to add any two vectors in the set and get another vector that's also in the set. Similarly, you must be able to multiply any vector by a scalar (a regular number) and get a vector that's still inside the set. Scalars usually come from a field, most commonly the set of real numbers, $\mathbb{R}$.

These operations must follow ten specific axioms, but they boil down to common-sense algebraic properties. For example, vector addition needs to be commutative ($u + v = v + u$) and associative ($u + (v + w) = (u + v) + w$). There must be a zero vector, which acts as an additive identity ($v + 0 = v$), and every vector must have an additive inverse ($-v$) such that $v + (-v) = 0$. Scalar multiplication has similar rules about distribution and association.

The real power of vector spaces comes from combining vectors. A linear combination is what we call an expression built from a set of vectors by multiplying each vector by a scalar and adding the results.

Here, $v_1, v_2, \dots, v_n$ are vectors and $c_1, c_2, \dots, c_n$ are scalars. The resulting vector, $w$, is a linear combination of the original vectors.

This leads to a crucial idea: the span. The span of a set of vectors is the collection of all possible linear combinations you can make from them. Think of it as every point you can possibly reach using only those vectors.

If you have a single vector in 3D space, its span is a line passing through the origin. If you have two vectors that don't point along the same line, their span is a plane. And if you have three vectors that don't all lie on the same plane, their span is all of 3D space.

Sometimes, a vector space can live inside a larger one. This is called a subspace. For a set of vectors to be a subspace, it must be a vector space in its own right and satisfy three conditions:

An easy example is a plane through the origin inside of 3D space. Any two vectors on that plane, when added, produce another vector on the same plane. Scaling a vector on the plane keeps it on the plane. And the zero vector is at the origin, which is on the plane.

A basis is a special set of vectors for a given space. Think of it as a minimal toolkit for building every other vector in that space. A basis must do two things: its vectors must span the entire space, and they must be linearly independent. Linear independence means that no vector in the basis can be created as a linear combination of the others. It's an efficient set—no redundant vectors.

For the 2D Cartesian plane, $\mathbb{R}^2$, a standard basis is the pair of vectors $v_1 = (1, 0)$ and $v_2 = (0, 1)$. They are linearly independent, and you can reach any point $(x, y)$ on the plane with the linear combination $x v_1 + y v_2$. Since there are two vectors in this basis, the dimension of $\mathbb{R}^2$ is two. For 3D space, $\mathbb{R}^3$, the dimension is three, and so on.

Understanding these foundational concepts—spaces, spans, and bases—is the key to unlocking the rest of linear algebra.