Vector Spaces, Modules, and Linear Algebra

Let’s take a little trip back in time to grade school mathematics. What is five apples plus three apples? Easy, the answer is eight apples. What about two oranges plus two oranges? The answer is four oranges. What about three apples plus two oranges? Wait, that question is literally “apples and oranges”! But we can still answer that question of course. Three apples plus two oranges is three apples and two oranges. Does that sound too easy? We ramp it up just a little bit: What is three apples and two oranges, plus one apple and five oranges? The answer is four apples and seven oranges. Even if we’re dealing with two objects we’re not supposed to mix together, we can still do mathematics with them, as long as we treat each object separately.

Such an idea can be treated with the concept of vector spaces. Another application of this concept is to quantities with magnitude and direction in physics, where the concept actually originated. Yet another application is to quantum mechanics, where things can be simultaneously on and off, or simultaneously pointing up and down, or simultaneously be in a whole bunch of different states we would never think of being capable of existing together simultaneously. But what, really, is a vector space?

We can think of vector spaces as sets of things that can be added to or subtracted from each other, or scaled up or scaled down, or combinations of all these. To make all these a little easier, we stay in the realm of what are called “finite dimensional” vector spaces, and we develop for this purpose a little notation. We go back to the example we set out at the start of this post, that of the apples and oranges. Say for example that we have three apples and two oranges. We will write this as

$\displaystyle \left(\begin{array}{c}3\\2\end{array}\right)$

Now, say we want to add to this quantity, one more apple and five oranges. We write

$\displaystyle \left(\begin{array}{c}3\\2\end{array}\right)+\left(\begin{array}{c}1\\5\end{array}\right)$

Of course this is easy to solve, and we have already done the calculation earlier. We have

$\displaystyle \left(\begin{array}{c}3\\2\end{array}\right)+\left(\begin{array}{c}1\\5\end{array}\right)=\left(\begin{array}{c}4\\7\end{array}\right)$

But we also said we can “scale” such a quantity. So suppose again that we have three apples and two oranges. If we were to double this quantity, what would we have? We would have six apples and four oranges. We write this operation as

$\displaystyle 2\left(\begin{array}{c}3\\2\end{array}\right)=\left(\begin{array}{c}6\\4\end{array}\right)$

We can also “scale down” such a quantity. Suppose we want to cut in half our amount of three apples and two oranges. We would have one and a half apples (or three halves of an apple) and one orange:

$\displaystyle \frac{1}{2}\left(\begin{array}{c}3\\2\end{array}\right)=\left(\begin{array}{c}\frac{3}{2}\\1\end{array}\right)$

We can also apply what we know of negative numbers – we can for example think of a negative amount of something as being like a “debt”. With this we can now add subtraction to the operations that we can do to vector spaces. For example, let us subtract from our quantity of three apples and two oranges the quantity of one apple and five oranges. We will be left with two apples and a “debt” of three oranges. We write

$\displaystyle \left(\begin{array}{c}3\\2\end{array}\right)-\left(\begin{array}{c}1\\5\end{array}\right)=\left(\begin{array}{c}2\\-3\end{array}\right)$

Finally, we can combine all these operations:

$\displaystyle 2\left(\left(\begin{array}{c}3\\2\end{array}\right)+\left(\begin{array}{c}1\\5\end{array}\right)\right)=2\left(\begin{array}{c}4\\7\end{array}\right)=\left(\begin{array}{c}8\\14\end{array}\right)$

For vector spaces, the “scaling” operation possesses a property analogous to the distributive property of multiplication over addition. So if we wanted to, we could also have performed the previous operation in another way, which gives the same answer:

$\displaystyle 2\left(\left(\begin{array}{c}3\\2\end{array}\right)+\left(\begin{array}{c}1\\5\end{array}\right)\right)=2\left(\begin{array}{c}3\\2\end{array}\right)+2\left(\begin{array}{c}1\\5\end{array}\right)=\left(\begin{array}{c}6\\4\end{array}\right)+\left(\begin{array}{c}2\\10\end{array}\right)=\left(\begin{array}{c}8\\14\end{array}\right)$

We can also apply this notation to problems in physics. Suppose a rigid object acted on by a force of one Newton to the north and another force of one Newton to the east. Then adopting a convention of Cartesian coordinates with the positive x-axis oriented towards the east, we can calculate the resultant force acting on the object as follows

$\displaystyle \left(\begin{array}{c}1\\0\end{array}\right)+\left(\begin{array}{c}0\\1\end{array}\right)=\left(\begin{array}{c}1\\1\end{array}\right)$

This is actually a force with a magnitude of around $1.414$ Newtons, with a direction pointing towards the northeast, but a discussion of such calculations will perhaps be best left for future posts. For now, we want to focus on the two important properties of vector spaces, its being closed under the operations of addition and multiplication by a scaling factor, or “scalar”.

In Rings, Fields, and Ideals, we discussed what it means for a set to be closed under certain operations. A vector space is therefore a set that is closed under addition among its own elements and under multiplication by a “scalar”, which is an element of a field, a concept we discussed in the same post linked to above. A set that is closed under addition among its own elements and multiplication by a scalar which is a ring instead of a field is called a module. Another concept we discussed in Rings, Fields, and Ideals and also in More on Ideals is the concept of an ideal. An ideal is a module which is also a subset of its ring of scalars.

Whenever we talk about sets, it is always important to also talk about the functions between such sets. A vector space (or a module) is just a set with special properties, namely closure under addition and scalar multiplication, therefore we want to talk about functions that are related to these properties. A linear transformation is a function between two vector spaces or modules that “respect” addition and scalar multiplication. Let $u$ and $v$ be any two elements of a vector space or a module, and let $a$ be any element of their field or ring of scalars. By the properties defining vector spaces and modules, $u+v$ and $av$ are also elements of the same vector space or module. A function $f$ between two vector spaces or modules is called a linear transformation if

$\displaystyle f(u+v)=f(u)+f(v)$

$\displaystyle f(av)=af(v)$

Linear transformations are related to the equation of a line in Cartesian geometry, and they give the study of vector spaces and modules its name, linear algebra. For certain types of vector spaces or modules, linear transformations can be represented by nifty little gadgets called matrices, which are rectangular arrays of elements of the field or ring of scalars. The vectors (elements of vector spaces) which we have featured in this post can be thought of as matrices with only a single column, or sometimes called column matrices. We will not discuss matrices in this post, although perhaps in the future we will; they can be found, along with many other deeper aspects of linear algebra, in most textbooks on linear algebra or abstract algebra such as Linear Algebra Done Right by Sheldon Axler or Algebra by Michael Artin.

References:

Vector Space in Wikipedia

Module on Wikipedia

Linear Algebra Done Right by Sheldon Axler

Algebra by Michael Artin