Subspaces

Let's say you have a set, U, of vectors and it is a subset of a set, V, of vectors (U ⊆ V). This situation is shown in the following diagram:

Figure 4.1 – The set U as a subset of V

Is it possible that U is a subspace of V? Well, yes. It has met the first condition for subspaces, namely, that the potential subspace has to be a subset of the bigger vector space's set of vectors. What's next? Well, U also has to be a vector space using the same field as the vector space V. This seems like it might be an exhaustive thing to do, but it has been proven that we only need to check for three small conditions to make sure U is a subspace of V, and two of them have to do with the closure property from Chapter 3, Foundations. As a reminder, here it is:

• Closure: For every a,b ∈ A, a ֎ b produces an element, c, that is also in the set A. f: A × A → A

Armed with the concept of a subset and closure...