Course Notes for Linear Algebra

In the definitions in Section 1.2, I defined vectors and Euclidean space in any dimension, not just the familiar two and three dimensions. If I want to work in seven dimensions, I simply need to write vectors with seven components. This is a strange and new thing. Conventional geometry deals with things I can draw and see, which are things that necessarily exist in two or three dimensions. Higher dimensions are something new and bizzare, but also fascinating. (Higher dimensions are not just a mathematical fascination: doing higher dimensional geometry is also extremely practical. I’ll try to provide justification for this practicality before the course is finished.)

In this section, I want to talk through some constructions that might help to build some intuition and familiarity with higher dimensions. Working in higher dimensions is a new kind of challenge, so buidling some intuition can be extremely valuable. I do need to use indirect (mostly algebriac)methods, since I can’t visualize higher dimensional spaces.

In some contexts, particularly in physics using Einstein’s general relativility, the fourth dimension is often treated as time. That’s not the approach in this course. I am interested in treating all these higher dimensions as environments for geometry. I want to imagine objects, paths, intersections, and many other geometric ideas in these higher dimensions. In the applications of linear algebra, higher dimensions can be given a huge variety of interpretation, including treating one dimension as time. Those applications can come afterwards, after the theory is well developed. In linear algebra, I treat all dimensions geometrically.

I’ll start building higher dimensional intituion by talking about how some familiar objects generalize into these higher dimensions.

In the plane, all points that are exactly one unit from the origin form the unit circle. In three dimensional space, all points that are exactly one unit from the origin form the unit sphere. This is the start of a pattern. Following this logic, I can construct ‘spheres’ in higher dimensions. In any \(\RR^n\text{,}\) all objects which are exactly one unit from the origin form the sphere in that dimension.

What do I know about these higher-dimensional spheres? First, I can look at their equations. In \(\RR^2\text{,}\) the circle has the equation \(x^2 + y^2 = 1\text{.}\) In \(\RR^3\) the sphere has equation \(x^2 + y^2 + z^2 = 1\text{.}\) The sphere in \(\RR^4\) has equation \(w^2 + x^2 + y^2 + z^2 = 1\text{.}\) The sphere in \(\RR^n\) has equation \(x_1^2 + x_2^2 + \ldots + x_n^2 = 1\text{.}\)

Getting this intuitive sense is fundamentally difficult. It is worth repeating that the major challenge of higher dimensions geometry is the fact that I can’t see it, since I am so used to seeing the geometry I am working with. However, there are still some indirect geometric ways of thinking that can help me understand higher-dimensional objects.

If I take the normal sphere in \(\RR^3\) and slice it vertically, each cross-section is a circle, as illustrated in Figure 1.4.1.. The cross-section at the widest point is a unit circle, and as I move along the \(y\) axis, the cross-sections are smaller and smaller circles. I can think of the sphere as an object whose slices are all circles (though it is not the unique object with that property). I could also think of the sphere as an infinite stack of infinitely thin circles, starting with small circles, growing to the the unit circles at the middle, and then shrinking again as I move further across the sphere.

This notion of slicing works in higher dimensions. The sphere in \(\RR^4\) has slices, which are usual three-dimensional spheres. That’s a bizarre idea, because trying to think in four dimensions is a bizarre task, and one that defies visualization. I can’t see an infinite stack of spheres all right next to each other. But this slicing into spheres is a useful way of thinking about the sphere in \(\RR^4\text{.}\) The sphere in four dimensions is an object whose slices are spheres in three dimensions; it can be thought of as infinitely many spheres stacked up. This is still strange: the stacking happens in a new direction, so that all points on adjacent spheres are right next to each other, just like the circles stack up to make the ordinary sphere are right next to each other. But maybe this helps us think a bit about what kind of object the sphere in \(\RR^4\) is. This pattern could continue: the five dimensional sphere has slices which are four dimensional spheres, and so on.

After the sphere, I want to consider square objects. To understand the higher-dimensional cubes, I want to talk about how to build each cube from lower-dimensional pieces. I’ll start at the very beginning, in \(\RR^1\text{,}\) which is just the number line.

The ‘cube’ in \(\RR\) is the interval \([0,1]\text{.}\) It can be defined by the inequality \(0 \leq x \leq 1\) for \(x \in \RR\text{.}\)

In \(\RR^2\text{,}\) the ‘cube’ is just the ordinary square. It is all vectors \(\begin{pmatrix} x \\ y \end{pmatrix}\) such that \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\text{.}\) The square can be formed by taking two intervals (1-dimensional cubes) and connecting the matching vertices. This will become the way in which I build higher-dimensional cubes: each cube is formed of two cubes from the previous dimension with all the matching vertices connected.

Now I’ll look at \(\RR^3\) and follow the construction process. If I take two squares and connect the matching vertices, I do indeed get the ordinary cube. One of the two squares is the front of the cube, and the other is the back (or top and bottom, left and right, depending on how you orient the pieces). To make this a bit more clear, in Figure 1.4.3, I’ve drawn the two original squares in black and the four connecting lines in red. I could also describe the cube (including its interior) as all points where the coordinates satisfy \(x \in [0,1]\text{,}\) \(y \in [0,1]\) and \(z \in [0,1]\text{.}\)

Then I can simply keep extending. In \(\RR^4\text{,}\) I will take two copies of the 3D cube and connect all the matching vertices. In Figure 1.4.4, I’ve shown the connecting lines again in red. This diagram is a poor picture, being a 2D representation of a 4D object, but it does give some idea of the construction of the shape. The 4D cube is often called the tesseract. I could again describe it as all vectors in \(\RR^4\) such that \(w \in [0,1]\text{,}\) \(x \in [0,1]\text{,}\) \(y \in [0,1]\) and \(z \in [0,1]\text{.}\)

There is a cube in each \(\RR^n\text{,}\) consisting of all vectors where all components are in the interval \([0,1]\text{.}\) Each can be constructed by joining two copies of a lower-dimensional \((n-1)\)-cube with edges between matching vertices. I haven’t tried to draw diagrams for any of these higher-dimensional cubes, since the diagrams quickly become very difficult to parse. But, hopefully, the diagram of the tesseract gives a sense of how to think about this un-seeable four dimensional object.

The cube is an example of a solid object with flat sides and sharp corners. Such objects are called polytopes (polygons in two dimensions). The study of polytopes in higher dimensions is itself a rich branch of mathematics with a long history.

In addition to the cubes, I want to introduce another kind of higher-dimensional polytope, called the cross-polytope. It generalizes the idea of a diamond shape in two and three dimensions. It also lets me show another way of trying to get a sense of what higher-dimensional objects look like.

In \(\RR^2\text{,}\) the cross-polytope is the diamond with vertices \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) , \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\text{,}\) \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) , and \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) . In each dimension, cross-polytopes are built by adding two vertices at \(\pm 1\) in the new axis direction. Then, edges are added connecting the two new vertices to each existing vertex (but the two new vertices are not connected to each other).

In \(\RR^3\text{,}\) if I take the diamond, add two points on either side and connect the new points to all four original vertices of the diamond, I am essentially building a square pyramid on either side of the diamond. The result is called the octahedron and has vertices \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\) , \(\begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix}\) , \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\text{,}\) \(\begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix}\) , \(\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\text{,}\) and \(\begin{pmatrix} 0 \\ 0 \\ -1\end{pmatrix}\text{.}\) In Figure 1.4.6, I’ve tried to draw the octahedron in perspective, to give a sense of it as a 3D object.

I can continue to build cross-polytopes in higher dimensions using this process: adding vertices on either side and connecting the new vertices to all previous vertices (but not to each other). If I choose the new vertices as points in the new axis direction, then all the vertices of the \(n\)-cross-polytope are vertices with \(\pm 1\) in one component and zero in all other components. Each vertex is connected to all other vertices except its opposite. I haven’t tried to draw diagrams of any of these higher cross-polytopes.

One way of understanding higher-dimensional objects (since I can’t directly see them) is to just draw their lower-dimensional components. For polyhedra, I can do this by only drawing their vertices and edges (and not trying to represent the higher-dimensional structures that include those vertices and edges). I’m going to do this for the cross-polytopes to demonstrate the idea. The diamond (2-cross-polytope) has four vertices with all connections except for opposite vertices.

The octahedron is the 3-cross polytope. I tried to draw a 3D image in perspective in Figure 1.4.6. However, if I don’t try to get a 3D perspective and just draw the vertices and edges, I get the vertex diagram instead of the entire object in perspective. I’ve shown this vertex diagram in Figure 1.4.8. It looks like a flattened version of the octahedron, which is precisely what I want it to do. Like the diamond, all vertices except for those directly opposite are connected.

The construction of the higher cross-polytopes, where I add two vertices and connect them to all existing vertices, consistently builds a set of vertices where each vertex is attached, by an edge, to all other vertices except for the one directly opposite it. If I collapse the vertices and edges onto a 2D diagram, that means I draw all the lines between our vertices except those directly opposite. The 4-cross polytope has \(8\) vertices connected in this way and its vertex diagram is shown in Figure 1.4.9.

Now I just keep going. In Figure 1.4.10, I show the 5-cross-polytope, whichs \(10\) vertices connected this way.

I can go as far as I want in this way. Each time I go up a dimension, I add two vertices. The 8-cross-polytope (in \(\RR^8\)) has \(16\) vertices connected this way, as shown in Figure 1.4.11.

To show how wild these vertex diagrams get, consider Figure 1.4.12, the vertex diagram of the 16-cross-polytope (in \(\RR^{16}\)), which has \(32\) vertices connected this way.

These images are obviously not full pictures of these higher dimensional objects, but just some kind of 2D shadows which give me some information and intuition for the objects. Everything in higher dimensions must be done indirectly, but working indirectly, I can produce some useful and interesting visualizations that let me start to understand the objects. Hopefully it is a little bit interesting to have some mental pictures of a 16 dimensional diamond-shaped prism, even if that mental picture is just of all the corners and edges collapsed down only a flat surface.