Today, in my leisure time, I want to talk about a very silly nonsense: the dimension of a base in algebra.
This concept is well known by all because it is simple (in principle) and used in daily life, therefore it is a concept that we have assumed.
Obviously and, above all, theoretically, a base is a generator of everything that contains a space. That is, if we consider a space (usually vectorial), a base must generate all the content, ALL, of said vector space.
And now, since you see that I am going back in the definition, it is convenient to know that it is a vector space.
A vector space, so cold, for anyone who thinks, would come to mind looking around and therefore think of a three-dimensional “space.” That is, I would think of volume.
The point is that a vector space is more than a volume space. A vector space itself is a group of elements (any) and an operation that can be done between them that, as a result, another element.
What can be a vector space?. Well, for example, as I put a long time ago, people dressed. It is a rare example but it is a vector space and is not seen from the world of mathematics. Another example, matrices; Matrices have a form of vector space since they are a group of elements to which you can have an operation (example, the sum) that gives elements of itself (another matrix) and an external operation (multiply by any number that does not Is an element of the vector space) than another matrix.
Another vector space, since the polynomials since a polynomial (of all life) we can decompose it according to its elements, read an element as in what can be decomposed, ie 1 + x + x ^ 2 has three elements, 1, The x and x ^ 2 then we can say that this polynomial has as values [1, 1, 1] concerning [1, x, x ^ 2] that would be the basis … and I have already put the concept of base.
A base is a generator of said space. That is to say, it is an element of the vectorial space from which and thanks to the external (and internal) operations of said space all the elements are generated. It sounds complicated, but it’s very simple. It is a series of elements of the vector space from which the rest depend on them. For example, of the example of the people with jersey, a base would be the colors of the different jerseys since, in this vectorial space, the base and the elements of the space totally agree. That is not the same a base with a generator system … but that is already a subject of mathematics and here … although using it, I try not to mix it.
And what about the dimension ?. For dimension is the number of elements that have the basis. It is a simple concept, very simple although some will make the penis a mess.
The fact is that vector spaces and bases are many … if we went down the street realizing what surrounds us and joining them to the concept of vector space and base, we would laugh a lot.
Regarding dimension, it has its properties, and, as I said before, we use them without realizing it. For example, we live in a world of (according to what string theory we take) up to 16 dimensions. For ordinary mortals, we see that we live in a vector space of 3 dimensions, high, wide and long although we sense that there are 4 dimensions if we take time to be a variance or something that varies.
We can also easily think of a generative system of this vector space (the one we live in) being, for example, 1 cm for each axis (which are 3, high, wide and long) and 1 second for time. That is, we can all position and draw it with those elements of space. But being a generating system does not mean that it is based since a group of elements that is part of the vector space and that can create all the vector space does not mean that it is base. As an example, if we take a paper (which is a vector space of 2 dimensions) there are many ways to take two elements (two lines) that, from them, we can create, place and place any element contained in said paper .. . what things. And that applies not only to that paper, but to any vector space, of course.
What we do not realize is that we have two problems with that way of thinking. The first is that there is always a smaller or different unit (for example, instead of centrimeters, we can see millimeters or feet or microns or Plank …) without knowing which is the minimum unit that generates all our vector space. The second is the reference, that is, measuring or locating things in reference to … your house, mine, the center of the Sun, at the head of a goose … and in this case, of What a goose ?. That is, problems of geometry of the whole life since they are based on the algebra that I have just told you. And all thanks to that, in part, we are seeing and living in a subspace of a much larger vector space, with its base changes, its definition of coordinates… a fun with flags (as Sheldon would say).
The fact is that now I understand why an old professor of physics of my career spent the day laughing. If I remember correctly I taught math, algebra to be exact. It was a guy like House, lame and with cane. He laughed continuously and nobody knew because he laughed while he walked … now I understand that he laughed because he saw vector spaces everywhere and their bases, which, deep down, now I understand is very funny.