Basis of Abstract Vector Space

What does the dimension of a vector space really mean? What does it mean for a set of vectors to span a vector space? What is a linear combination of vectors for that matter? In this article I will try and give you and intuitive understanding of each of these concepts as I progress towards the definition of a basis of a vector space. I will include some example as well because I believe those are the best way to build an intuition. If you don’t yet have an understanding of a vector space see my other article here, and there are also many resources online.

Linear Combination of Vectors

Let’s get started! A linear combination of a set of vectors in some vectors space V over a field F is of the following form:

where each scalar a if from the field F and each vector v is from the vector space V. We know that from closure properties of vector addition and scalar multiplication over a vector space that this new linear combination must also be in the vector space as well.

Span and Spanning Sets

The span of a set of vectors is all of the possible linear combinations of these vectors. ie Varying the coefficient a ∈ F such every combination is achieved.

Two example that are familiar to use are a line in 2D and a plane in 3D.

The line and plane can each be seen as all of the linear combinations of two respective sets of vectors. We can see that the line need only be defined by one vectors, as scalar multiplication sweeps out the entire line. Somehow we can tell that one vector is not enough to define a plane. We will have the language to articulately distinguish these differences by the end of the article. Note that in each of these cases closure holds. Also note that each of these subspaces intersect the origin. In fact its true that they must intersect the origin, you can justify this by thinking back to the construction of a vector space, of my using some scalar multiplication :).

We say that a set of vectors, S = {v1, …vn} where v1, …vn ∈ V, spans a vector space V, if the span of those vectors is equal to V. In other words if every vector v ∈ V is expressible as some linear combination of the vectors in S. This is clearest through some examples. The simplest being the line through the origin above. It clear that if you take one vector and stretch it forward and backward that will span the entire line.

Linear Independence

One more defintion and then we will be able to understand a basis! This final definition is that of linear independence. I think the that formal defintion can be somewhat confusing so I will try and explain this geometrically and intuitively instead. We say that a set of vectors are linearly independent if each vector in the set cannot be formed by a linear combination of the rest of the vectors in the set. In other words there is no way to add or scale the remaining vectors to equal the vector of interest. What this means is that each vector in the set bring with it essential new information, that is not captured in the span of the remaining vectors.

Basis

Now we are ready to define a basis. We say that a set of vectors in V {v1…vn} is a basis for V if they span V and are linearly independent. Now what does this mean? First thing to note is that the basis is not unique. A simple example is that (1,1) and (2,2) could both be the same basis of a space because the are in each other spans. What is not completely obvious, but may be intuitive is that fact that although basis’ are not unique, they all must have the same number of elements. The number of elements in a basis is called the dimension of the space. The proof isn’t too difficult but you can convince yourself of this by realizing that in a basis each new vector must add new information in order to be linearly independent from the previous. So in one sense since the vector space is describable by some minimal amount of information, it makes sense that it two sets of vectors satisfy the requirements for a basis, they would hold the same information and have the same number of vectors.

Note

This post is becoming a little longer than expected so I will hold most of the interesting example for the next one. I will leave you to think about whether these definitions math the intuition you already have. Does it make sense that we call the cartesian plane, R², 2-dimensional (ie that its basis contains 2 elements)? Why isn’t it possible to describe the plan with just one vector, why is three vectors redundant? What is a common basis used that you learning in grade school?