Linear Combinations, Span, and Finite vs. Infinite Dimensional Vector Spaces

Let \(v_1, \dots, v_m\) be some vectors in a vector space \(V\). Now \(V\) is closed under addition and scalar multiplication. So, any combination of these vectors built by applying these operations of addition and scalar multiplication is also a vector in \(V\).

For example, if \(a,b\) are elements of the field over which \(V\) is a vector space, \( a v_1 + b v_3 \) is also a vector in \(V\). Such things are called linear combinations of the vectors \(v_1, \dots, v_m \).

Let \(V\) be a vector space over a field \(\mathbb{F}\). A linear combination of vectors \(v_1, \dots, v_m \) is a vector \(v\) defined by some choice of \(m\) field elements \(a_1, \dots, a_m \in \mathbb{F}\) as \( v = a_1 v_1 + \cdots a_m v_m = \sum_{i=1}^m a_i v_i \)
Thinking back to our defintion of , it should be clear that the set of all such linear combinations of \(v_1, \dots, v_n\) is a subspace of \(V\). We call this subspace of linear combinations the span of \(v_1, \dots, v_n\).
A subset \(U\) of a vector space \(V\) is a subspace of \(V\) if \(U\) is also a vector space over the same field, using the same addition and scalar multiplication rules.

We sometimes write span\( (v_1, \dots, v_m) \) to denote the span of the list \( v_1, \dots, v_m \). We also say \(v_1, \dots, v_m\) spans \(V\).

We don't have enough tools yet to define the dimension of a vector space, but suprisingly we can already distinguish finite-dimensional vector spaces from infinite-dimensional ones. One boring subspace of any vector space \( V \) is itself, and we can ask what vectors are needed to span it.

If some finite list of vectors \( v_1, \dots, v_m \) spans \(V\), we say \(V\) is finite-dimensional. If it is impossible for any finite list of vectors to span \(V\), we say \(V\) is infinite dimensional.

So far we have been talking about finite lists of vectors. What if we have a countably infinite list \( v_1, v_2, \dots \), or even an uncountable collection of vectors (say, all the vectors in some ball in \( \mathbb{R}^2 \) ). The concept of span still makes sense, but we have to be careful about what sort of linear combinations we allow.

We want to exclude linear combinations that use infinitely many terms, because this leads to a host of other problems, limits, and so on and requires analysis to think about. Instead we only allow finite linear combinations. That is, given a set of vectors, we only get to choose finitely many vectors to recombine to obtain a new one. Equivalently, only finitely many coefficients in the sum should be nonzero.

Let \(V\) be a vector space over a field \(\mathbb{F}\), and \(S\) some subset of vectors in \(V\). A linear combination of vectors in \(S\) is a choice of finitely many vectors \(v_1, \dots, v_m \) in \(S\), together with a choice of \(m\) field elements \(a_1, \dots, a_m \in \mathbb{F}\), to obtain a new vector \(v\): \( v = a_1 v_1 + \cdots a_m v_m = \sum_{i=1}^m a_i v_i \)

So now we can define the span of an arbitrary set of vectors.

Let \(V\) be a vector space over a field \(\mathbb{F}\), and \(S\) some subset of vectors in \(V\). The span of \(S\) is the set of all linear combinations of vectors in \(S\).

Most of the time, we will only consider spans of finite lists of vectors.

Consider the vector space \(\mathbb{R}^2\)

  • What is the span of the unit ball of vectors around the point (2,2)?
  • Can you think of an uncountable set of vector in \(\mathbb{R}^2\) whose span is a strict subspace of \(\mathbb{R}^2\)?