In Vector Spaces, Modules, and Linear Algebra we learned about vectors, and defined them as elements of a set that is closed under addition and scalar multiplication. This is a pretty abstract concept, and in that post we used an example of “apples and oranges” to express it. However we also mentioned that many other things are vectors; for instance, states in quantum mechanics, and quantities with a magnitude and direction, such as forces. It is these quantities with a magnitude and direction that we will focus on in this post.
We will use the language that we developed in Matrices in order to make things more concrete. We will focus on two dimensions only in this post, in order to simplify things, although it will not be difficult to generalize to higher dimensions. We develop first a convention. The vector
represents a quantity with magnitude “” (meter, or meter per second, or Newton, etc.) going to the right (or east). Similarly, the vector
represents a quantity with magnitude going to the left (or west). Meanwhile, the vector
represents a quantity with magnitude going upward (or to the north). Finally, the vector
represents a quantity with magnitude going downward (or to the south). These vectors we have enumerated all have magnitude , therefore they are also called unit vectors. Since they are vectors, we can “scale” them or add or subtract them from each other to form new vectors. For example, we can “double” the upward-pointing unit vector,
to obtain a vector again pointing upward but with a magnitude of . We can also “add” the right-pointing unit vector to the upward-pointing unit vector, as follows:
We can easily infer that this vector will point “diagonally” upward and to the right (or to the northwest). But what will be its magnitude? For this we introduce the concept of the transpose. The transpose of a matrix is just another matrix but with its rows and columns interchanged. For a column matrix, we have only one column, so its transpose is a matrix with only one row, as follows:
Now, to take the magnitude of a vector, we take the square root of the product of the transpose of a vector and the vector itself. Note that the multiplication of matrices is not commutative, so it is important that the row matrix be on the left and the column matrix (the vector) be on the right. It is the only way we will obtain an ordinary number from the matrices.
Applying the rules of matrix multiplication, we see that for a vector
the magnitude will be given by the square root of
This should be reminiscent of the Pythagorean theorem. As we have already seen in From Pythagoras to Einstein, this ancient theorem always shows up in many aspects of modern mathematics and physics. Going back to our example of the vector
we can now compute for its magnitude. Multiplying the transpose of this vector and the vector itself, in the proper order, we obtain
and taking the square root of this number, we see that the magnitude of our vector is equal to .
In Matrices we mentioned that a square matrix may be used to describe linear transformations between vectors. Now that we have used the language of vectors to describe quantities with magnitude and direction, we also show a very special kind of linear transformation – one that sends a vector to another vector with the same value of the magnitude, but “rotated” or “reflected”, i.e. with a different direction. We may say that this linear transformation describes the “operation” of rotation or reflection. This analogy is the reason why linear transformations from a vector space to itself are also often referred to as linear operators, especially in quantum mechanics.
We make this idea clearer with an explicit example. Consider the matrix
We look at its effect on some vectors:
From these basic examples one may infer that our matrix represents a counterclockwise “rotation” of ninety degrees. The reader is encouraged to visualize (or better yet draw) how this is so. In fact, we can express a counterclockwise rotation of any angle using the matrix
We consider next another matrix, given by
We likewise look at its effect on some vectors:
What we see now is that this matrix represents a “reflection” along the horizontal axis. Any reflection along a line specified by an angle of is represented by the matrix
The matrices representing rotations and reflections form a group (see Groups) called the orthogonal group. Since we are only looking at rotations in the plane, i.e. in two dimensions, it is also more properly referred to as the orthogonal group in dimension , written . The matrices representing rotations form a subgroup (a subset of a group that is itself also a group) of the orthogonal group in dimension , called the special orthogonal group in dimension and written .
The reader is encouraged to review the concept of a group as discussed in Groups, but intuitively what this means is that by multiplying two matrices, for instance, representing counterclockwise rotations of angles and , then we will get a matrix which represents a counterclockwise rotation of angle . In other words, we can “compose” rotations; and the composition is associative, possesses an “identity” (a rotation of zero degrees) and for every counterclockwise rotation of angle there is an “inverse” (a clockwise rotation of angle , which is also represented as a counterclockwise rotation of angle ).
It can be a fun exercise to derive this equation using the laws of matrix multiplication and the addition formulas for the sine and cosine functions from basic trigonometry.
This is what it means for , the matrices representing rotations, to form a group. Reflections can also be considered in addition to rotations, and reflections and rotations can be composed with each other. This is what it means for , the matrices representing rotations and reflections, to form a group. The matrices representing reflections alone do not form a group however, since the composition of two reflections is not a reflection, but a rotation.
Technically, the distinction between the matrices representing rotations and the matrices representing reflections can be seen by examining the determinant, which is a concept we will leave to the references for now.
It is worth repeating how we defined the orthogonal group technically – it is the group of matrices that preserve the magnitudes of vectors. This gives us some intuition as to why they are so special. There are other equivalent definitions of . For example, they can also be defined as the matrices which satisfy the equation
where the matrix is the transpose of the matrix , which is given by interchanging the rows and the columns of , as discussed earlier, and
is the “identity” matrix, which multiplied to any other matrix (on either side) just gives back . This may also be expressed by saying that the group is made up of the matrices whose transpose is also its inverse (and vice versa).
In summary, we have shown in this post one specific aspect of vector spaces and linear transformations between vector spaces, and “fleshed out” the rather skeletal framework of sets that are closed under addition and scalar multiplication, and functions that respect this structure. It is important to note of course, that the applications of vector spaces and linear transformations are by no means limited to describing quantities with magnitude and direction.
Another concept that we have “fleshed out” in this post is the concept of groups, which we have only treated rather abstractly in Groups. We have also been using the concept of groups in algebraic topology, in particular homotopy groups in Homotopy Theory and homology groups and cohomology groups in Homology and Cohomology, but it is perhaps the example of the orthogonal group, or even better the special orthogonal group, where we have intuitive and concrete examples of the concept. Rotations can be composed, the composition is associative, there exists an “identity”, and there exists an “inverse” for every element. The same holds for rotations and reflections together.
These two subjects that we have discussed in this post, namely linear algebra and group theory, are in fact closely related. The subject that studies these two subjects in relation to one another is called representation theory, and it is a very important part of modern mathematics.
Algebra by Michael Artin