Vector Spans

Definition

The span of vectors v1,v2,…,vnv_1, v_2, \dots, v_n from a Vector Space V\mathbb{V} is the set of all possible Linear Combinations of these vectors.

Formally:

span{v1,…,vn}={c1v1+c2v2+⋯+cnvn:ci∈F}\text{span}\{v_1,\dots,v_n\} = \{c_1v_1 + c_2v_2 + \dots + c_nv_n : c_i \in \mathbb{F}\}

where F\mathbb{F} is our scalar field (usually R\mathbb{R} or C\mathbb{C}).

Examples

In R2\mathbb{R}^2

Consider v=(10)v = \begin{pmatrix} 1 \ 0 \end{pmatrix}

Then span{v}\text{span}{v} is the x-axis - all vectors of the form (c0)\begin{pmatrix} c \ 0 \end{pmatrix} where c∈Rc \in \mathbb{R}.

In Polynomial Space

Consider p(x)=1+xp(x) = 1 + x

Then span{p(x)}\text{span}{p(x)} consists of all polynomials of the form c(1+x)c(1 + x) where c∈Rc \in \mathbb{R}.

Properties

  1. The span is always a Subspace
  2. The span is the smallest subspace containing the vectors
  3. span{0}={0}\text{span}{\mathbf{0}} = {\mathbf{0}}
  4. If vv is in the span of some vectors, adding vv to those vectors doesn't change the span

Why Do We Care?

Spans help us understand:

  1. How to generate subspaces
  2. Which vectors we can "reach" from our starting vectors
  3. How many vectors we need to describe a space (Dimension)
  4. Whether vectors are Linearly Dependent

Think of span as the "reach" of your vectors - it tells you all possible destinations you can get to using only linear combinations as your means of travel.

Exercise

Show that span{v1,v2}=span{v1}\text{span}{v_1,v_2} = \text{span}{v_1} if and only if v2v_2 is a scalar multiple of v1v_1.