Vector Spaces

Entry conditions

You should know what a field is and be comfortable with groups.

Definitions

A vector space over a field $F$ is a set $V$ equipped with two operations — vector addition and scalar multiplication — satisfying:

1. $(V, +, \mathbf{0})$ is an abelian group.
2. Scalar multiplication $F \times V \to V$ satisfies: $1 \cdot \mathbf{v} = \mathbf{v}$, $a(b\mathbf{v}) = (ab)\mathbf{v}$, $(a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$, and $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$.

A subspace is a subset of $V$ that is itself a vector space under the same operations.

A set of vectors $\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}$ is linearly independent if no nontrivial linear combination $a_1 \mathbf{v}_1 + \cdots + a_n \mathbf{v}_n = \mathbf{0}$ exists (i.e., all $a_i = 0$ is the only solution). A basis is a linearly independent set that spans $V$. The number of vectors in any basis is the dimension of $V$.

Vocabulary (plain language)

• Vector space: a set where you can add elements and scale them by numbers from a field.
• Basis: a minimal spanning set — the “coordinates” of the space.
• Dimension: how many basis vectors you need.

Intuition

The plane $\mathbb{R}^2$ is a vector space over $\mathbb{R}$. You can add arrows and stretch them. The standard basis $\{(1,0), (0,1)\}$ lets you express every vector as a linear combination. Dimension is $2$ because you need exactly two basis vectors.

Vector spaces abstract this: any set where addition and scaling obey the right rules is a vector space, regardless of what the “vectors” actually are. Polynomials of degree at most $n$ form a vector space of dimension $n + 1$.

Common mistakes

• Confusing a vector space with $\mathbb{R}^n$. Many vector spaces are not collections of columns of real numbers.
• Forgetting that the zero vector must be in every subspace.