Vector Spaces
Vectors: Magnitude and directions. You have probably seen a vector in a 2d plane with a line showing the magnitude and an arrow pointing in a direction. In mathematics, however, vectors expand beyond 2d space. When a mathematician is increasing the dimensional space they work on, they add one more coordinate to define each vector. For example, with a 2d space, only two coordinates are required to properly locate the post. In 7-dimensional space, however, 7 coordinates would be required to define each vector. Of course, defining larger vectors is only helpful if each different dimension has the same properties. The main properties are that addition, commutativity, inverses, and associativity. Another property is that there is a 0 vector where 0 added to the vector maintains the direction and magnitude of the vector.
Another main idea is vectors, and how vector spaces to work, is that of scalars. Scalars are simply numbers that multiply into each vector and change the vector’s magnitude. These scalars are not just limited to the real world. Rather, complex scalars become increasingly useful when continuing to look at vector spaces. I will try to cover complex scalars and properties in a future post. We have started using them in my own math class, and they are becoming increasingly useful for what seemed impossible.
Each of the above properties, along with how vectors increase in dimensional leads to the definition of what a vector space actually is. A vector space is a collection of vectors and operations that can be true for all vectors and all real numbers. The operations are just the same as those listed above. In total, there are 10 vector properties that must hold true.
Vector spaces are most commonly represented in matrices with defining conditions. The simplest example is the {0} space. All of the 10 above conditions are true for the 0 space; therefore, it is a valid vector space.
An example of what is NOT a vector space is below:
Notice how if you multiply the vector space by -1, you obtain a point that is not in the vector space, and therefore the matrix is not actually defining a vector space. This is shown by the red dot and arrow.
Another example that is not as obvious is the below matrix:
Again this is not a vector space since not all properties hold. See the two red vectors in the shaded region defined by the matrix. If these two vectors that ARE in the space are added, you obtain a vector that is NOT in the space. This means that again this is not a vector space.
Now for the fun: geometries in vector spaces. My math class has started to use vectors to prove or redefine geometric theorems. For example, see the drawing below:
This is a very elementary example of how similar triangles work. There are four main vectors making up this space represented in red. In blue, are vectors (b - a) and vector (d - c). Notice how those two resultant vectors are parallel to each other. This forms two different “triangles “ where each angle is the same. We can easily prove each angle using dot products of different vectors. Once proved, this means there is some common ratio between each of the side lengths. That also means that the vector a-b can be scaled to obtain the vector c-d. And, it means that vector a times a scalar is also vector c.
Another cool way to create a vector space is through a polynomial. The degree of a polynomial vector space with a finite polynomial degree is a space of n + 1. This is simply because it fits all of the criteria first listed above. It can be scaled and added. Just because it is in a different form does not exclude it from being a vector space.
One example of a defined vector space is the Minkowski space. The online definition claims that the Minkowski space is a “combination of three-dimensional Euclidean space and time into a four-dimensional manifold.” There are some other defining features that I do not really understand; however, this is an example of how a vector space can be abstract while still defining key physics principles.
References
Beardon, A. F. (n.d.). Vector Spaces. Algebra and Geometry, 102-134. doi:10.1017/cbo9780511800436.008
Minkowski space. (2022, December 04). Retrieved December 19, 2022, from https://en.wikipedia.org/wiki/Minkowski_space
SocraticaStudios (Director). (2016, October 20). [Video file]. Retrieved December 15, 2022, from https://www.youtube.com/watch?v=ozwodzD5bJM
Thetrevtutor (Director). (2016, May 05). [Video file]. Retrieved December 15, 2022, from https://www.youtube.com/watch?v=XDvSsDsLVLs