We've talked about vectors as arrows with magnitude and direction. But in linear algebra, the idea of a vector is much broader. A vector can be an arrow, a list of numbers, a polynomial, or even a function. What makes them all 'vectors' is that they belong to a special collection called a vector space.

A vector space is essentially a playground for vectors. It's a set of objects, which we call vectors, that follow a specific set of rules for two basic operations: vector addition and scalar multiplication. As long as these rules are obeyed, we can call the set a vector space.

Think of it like a sandbox. You can add sand together (vector addition) or take a certain amount of sand (scalar multiplication), and you're still left with sand in the sandbox. The rules of a vector space ensure that when you combine vectors or scale them, you always end up with another vector that's still inside the same space. It's a closed, consistent system.

Here are the ten official rules, or axioms, that a set must satisfy to be a vector space. Let $\vec{u}$, $\vec{v}$, and $\vec{w}$ be vectors in the space, and let $c$ and $d$ be scalars:

1. Closure under addition: If you add any two vectors $\vec{u}$ and $\vec{v}$, their sum $\vec{u} + \vec{v}$ is also in the space.
2. Commutativity of addition: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$. The order doesn't matter.
3. Associativity of addition: $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$. Grouping doesn't matter.
4. Zero vector: The space contains a zero vector $\vec{0}$ such that $\vec{v} + \vec{0} = \vec{v}$ for any vector $\vec{v}$.
5. Additive inverse: For every vector $\vec{v}$, there is an inverse vector $-\vec{v}$ such that $\vec{v} + (-\vec{v}) = \vec{0}$.
6. Closure under scalar multiplication: If you multiply any vector $\vec{v}$ by any scalar $c$, the result $c\vec{v}$ is also in the space.
7. Distributivity (vector sum): $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$.
8. Distributivity (scalar sum): $(c + d)\vec{v} = c\vec{v} + d\vec{v}$.
9. Associativity of scalar multiplication: $c(d\vec{v}) = (cd)\vec{v}$.
10. Scalar identity: $1\vec{v} = \vec{v}$.

The familiar 2D plane, known as $\mathbb{R}^2$, is a classic vector space. The vectors are points like $(x, y)$, and they follow all ten rules. The same is true for 3D space, $\mathbb{R}^3$, and even higher-dimensional spaces like $\mathbb{R}^n$.

But the concept is more powerful than that. Consider the set of all polynomials of degree 2 or less. A typical element, or 'vector', in this space looks like $p(x) = ax^2 + bx + c$. Let's check a couple of rules.

If we add two such polynomials, say $(ax^2 + bx + c)$ and $(dx^2 + ex + f)$, we get $(a+d)x^2 + (b+e)x + (c+f)$, which is still a polynomial of degree 2 or less. So, it's closed under addition. If we multiply a polynomial by a scalar $k$, we get $k(ax^2 + bx + c) = (ka)x^2 + (kb)x + (kc)$, which is also in the set. It turns out this set of polynomials satisfies all ten axioms, making it a perfectly valid vector space.

Within a vector space, we can create new vectors by stretching and adding existing ones. This process is called making a linear combination.

For example, if we have two vectors $\vec{v}_1$ and $\vec{v}_2$, a linear combination of them would be any vector $\vec{w}$ that can be written as:

Here, $c_1$ and $c_2$ are just any scalars. The set of all possible linear combinations of a group of vectors is called their span. The span is the entire region of the vector space you can reach using only those starting vectors.

If you have two non-parallel vectors in $\mathbb{R}^3$, their span is a plane passing through the origin. If you have just one non-zero vector, its span is a line passing through the origin. The span of a set of vectors is always a vector space itself.

This leads to a crucial idea: a subspace. A subspace is a vector space that is contained within another, larger vector space. Think of the 3D world we live in ($\mathbb{R}^3$). A flat plane that passes through the origin is a subspace of $\mathbb{R}^3$. Any line passing through the origin is also a subspace. They are smaller, self-contained vector spaces living inside the bigger one.

To be a subspace, a set must satisfy three simple conditions:

1. It must contain the zero vector.
2. It must be closed under vector addition. (If you add two vectors from the subspace, the result is still in the subspace.)
3. It must be closed under scalar multiplication. (If you scale a vector from the subspace, the result is still in the subspace.)

Why are subspaces important? They help us break down complex vector spaces into simpler, more manageable parts. Understanding the structure of a vector space often means understanding its various subspaces.

Vector spaces and subspaces are the fundamental building blocks for almost everything else in linear algebra. They provide the context for operations with matrices and understanding linear systems.