Mathematical Scholars Calculus 3

Having concluded our reviews and introduced some new arithmetic, we now proceed to explore a famous structure in linear algebra - vector spaces. While many of you will go on to study these spaces over an arbitrary field, we will restrict our attention in this course to real and complex vector spaces. We will call this set of numbers \(K\) (so that \(K\) is either \(\mathbb{R}\) or \(\mathbb{C}\)). Let’s begin with the abstract definition.

An extremely common alternative notation to using bold letters \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) for vectors is to place an arrow above them as in \(\vec{u}, \vec{v}, \vec{w}\text{.}\) Having now seen the abstract definition, let us gather several examples.

One may ask why column vectors are so popular and there are a couple of very reasonable responses to this question. One is that we can compute very effectively with column vectors (as we will see for the remainder of our lives). Another is that every finite dimensional vector space over \(K\) is in a very precise sense equivalent to a space of column vectors. This second response is important, but it comes with a caveat which we will discuss when the time comes!

Although we do, Sage is built not to care about the difference between row and column vectors. It uses the following syntax to define a vector and will interpret it as a column or a row, depending on how you operate on it.

All of the examples above share the property of being finite dimensional (which we will define soon). However, there are a great many vector spaces that do not have this property and are extensively studied.

There are a huge number of variants of Example 2.2.7 such as functions that are differentiable, bounded functions, functions that are integrable, and so on. The study of such vector spaces is one of the primary topics of functional analysis and we will only lightly dabble in this so as not to spoil the wonders that await!

One can also consider infinite column vectors or matrices, but we will stop this example madness here before it gets any further out of control. Instead, we give ourselves a bit more vocabulary concerning vectors and vector spaces.

Often we will refer to a vector subspace \(W\) as a linear subspace. This should come though with a big warning: with this vocabulary, not all lines are linear!

In order to verify that a given subset \(W\) of \(V\) is in fact a vector space, one needs to check that it is closed under addition and scalar multiplication. In other words, we need to see that if \(\mb{w}_0, \mb{w}_1\) are in \(W\) and \(\lambda\) is in \(K\text{,}\) then \(\mb{w}_0 + \mb{w}_1\) and \(\lambda \mb{w}_0\) are also in \(W\) (not just in \(V\)). We record this as a proposition.

Linear combinations and span are essential vocabulary when communicating how vectors sweep out spaces. For example, a linear combination of a single non-zero vector \(\mb{v}\) in \(\mathbb{R}^3\) is just a scaling of \(\mb{v}\) and its span is the set of all such scale multiples; namely, a line through the origin in the direction of \(\mb{v}\text{.}\) On the other hand, taking the span of two vectors \(\mb{v}_1, \mb{v}_2\) which are not on the same line will give a plane through the origin. Moreover, if we remember these vectors we get coordinates on this plane by taking the point \((x,y)\) to the linear combination \(x \mb{v}_1 + y \mb{v}_2\text{.}\)

Sage has a class that is a vector space or subspace of another vector space. However, Sage (like us) cares about what \(K\) is for such a space. Often it is preferable to use the rational numbers \(K = \mathbb{Q}\) which Sage writes as QQ. If you evaluate the following snippet, Sage will output a vector subspace which is the span of the two vectors, along with some information about the subspace that is defined. We will learn about these terms in the coming sections.

As you will show in the exercises, the span of any non-empty subset of a vector space is a vector subspace. However, some sets of vectors are better than others when it comes to spans. For example, the span of the set of all vectors in \(\mathbb{R}^2\) is \(\mathbb{R}^2\text{,}\) but that is not saying much. On the other hand, the span of the two standard basis vectors \(\mb{e}_1, \mb{e}_2\) is also \(\mathbb{R}^2\text{,}\) yet for these vectors we get a property that we didn’t have before.

As one is often interested in whether a given set of vectors \(\{v_1, \ldots, v_m\}\) in \(K^n\) is a basis, we will assert unambiguously that if there is any hope of this being true, we must have \(m = n\) (this will be proven later). Even then, it is not always the case that we have a basis. However, Sage can be used to detect whether a give set of row or column vectors \(\{v_1, \ldots, v_n\}\) in \(K^n\) is a basis. Indeed, one can simply define the vectors, take the span and check that the resulting subspace has dimension \(n\text{.}\) Why this works should be rather mysterious to the student at this point, but will be clarified in the coming sections.

A couple of comments on this definition are in order. First, we take a basis to be an ordered set of vectors, but often we don’t mention the ordering when it plays no role. It is not universally defined this way and occasionally is defined as just a set of vectors satisfying the uniqueness property and equation (2.2.4). Second, we defined a finite basis above, but one can also consider infinite ordered sets that are bases (‘bases’ is the plural of basis). However, in the infinite case there are more important variants to the above definition of basis involving infinite sums, so for now we will stick with the finite case.

On first exposure to the concept of a basis, a student may get the wildly incorrect idea that they are in short supply. In fact, there are infinitely many bases of any non-zero rational, real or complex vector space. Nonetheless, the key advantage to having a basis around is that you can label your vectors unambiguously with arrays of numbers and thus, you can compute!

To be clear, the abstract notion of a basis is that given in the definition. The computational perspective is that one can uniquely assign arrays of numbers to vectors and, as we will see, add and scalar multiply in an easy fashion. Finally, the geometric perspective is rooted in Descartes original insight that points in space could be given coordinates. This last perspective is extremely subtle in that the word ‘could’ has great importance. I will make the point here that a vector space is a space of vectors or points that does not come with a preferred set of coordinates (unless the space you start with is \(K^n\)). Once a basis is given, or chosen, we can then assign coordinates. Indeed, while we do not walk around with a three dimensional grid attached to our head, the space we live in still exists! I may tell you that `the coffee cup is on the table over there’, but you will not hear me say ‘the coffee cup is at \((-3, 2, 1)\)’.