Vector Spaces

We start with a non-empty set \(V\) of elements. A vector space over a field \(K\) is a set equipped with addition and scalar multiplication satisfying the usual axioms. In practice, a quick test for many common examples is to check that \(V\) is closed under (i) addition and (ii) scalar multiplication by elements of \(K\).

Terminology. A set is closed under an operation if performing the operation on elements of the set produces an element still in the set. Saying \(V\) is “over” a field \(K\) means scalar multiplication \(K\times V\to V\) is defined. Typical fields are the reals \(\mathbb{R}\) and complexes \(\mathbb{C}\).

Elements of a vector space are called vectors. Vector spaces arise from numbers, functions, sequences, polynomials, and more.

Subspaces

A subspace of a vector space is a subset that is itself a vector space (using the same operations). In short, it is a vector space contained in another vector space.

Span

Let \(T=\{v_1,v_2,\dots,v_m\}\). The span of \(T\) is the set of all finite linear combinations of vectors from \(T\):

\operatorname{span}(T)=\Big\{\, \sum_{i=1}^{m} c_i v_i : c_i\in K\,\Big\}.

The span of any set of vectors is a subspace. If one vector in \(T\) is a linear combination of the others (e.g., \(v_3=v_1+v_2\)), then removing it does not change the span. Thus, sometimes more vectors do not enlarge the span.

Linear Independence

A set \(T=\{v_1,\dots,v_n\}\) is linearly independent if the only solution to

c_1 v_1 + c_2 v_2 + \cdots + c_n v_n = 0

is the trivial one, \(c_1=\cdots=c_n=0\). Otherwise it is linearly dependent. Equivalently, if there exist coefficients, not all zero, with the sum equal to \(0\), then some vector (say \(v_1\)) can be written as a linear combination of the others:

v_1 = -\frac{c_2}{c_1}v_2 - \cdots - \frac{c_n}{c_1}v_n \quad (c_1\neq 0).

Such a vector is redundant for spanning, so the span is unchanged by removing it.

Minimal Spanning Set

A minimal spanning set of a vector space is a smallest set of vectors whose span equals the whole space. Equivalently, it is a spanning set that is linearly independent.

Basis of Vector Spaces

A basis of a vector space is a linearly independent set that spans the space. In finite dimensions, every vector can be written uniquely as a finite linear combination of basis vectors, and the number of basis elements is the dimension.

In infinite-dimensional spaces, linear combinations may require convergence notions (topology). These are addressed in notes on topological vector spaces.