@fresh_42
Let me explain myself. Of course, I did not mean the inner product operation and matrix multiplication to be the same.
Let's say we have two vectors (vector and covector to be precise) that we will regard as matrices with one row and one column. So, by definition of inner product we...
@fresh_42
I am afraid I am missing your point. We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of...
It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations:
1) matrix addition;
2) multiplication by the undelying field elements;
3) matrix multiplication.
Is the last one...
Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...
@fresh_42 Will it be correct (albeit not quite rigourous, but for now I am trying to grasp the very idea) to say that transformations are automorphisms of space (set, manifold) arising in context of consideration of general linear group and subgroups thereof?
Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably.
By «transformation» I mean mapping from some set...
Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses...
Not quite sure I understand how cosets can be groups themselves. They lack identity element, since it is already included in the generating subgroup (normal or non-normal).
Hello!
As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient.
Yes, the bundle of cosets in this case will be...
Thanks everyone for replies. The blurred picture seems to get good deal of clarity now. Just to be sure I got your ideas I try express them in less rigorous terms, please, let me know if I err.
The term «field» may refer to two (at least two) different concepts: set-based structure from abstact...
I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor...