Recent content by nomadreid

Thanks, Gavran. Your post explains the confusion very well and concisely, and clears up a whole lot of my confusion. Super! As well, wrobel, I have been reading the book by Halmos that you recommended, and this also clears up a lot of my previous confusion. So many thanks for the recommendation.

OK, downloaded.

Yes

Ah, I guess I mean a non-orthonormal vector space; i,e, a vector space in which not all of the basis vectors are orthogonal to one another.

I am older/less young than the poster, and have been receiving answers from the wonderful people on this forum for many years for my inexpert dabbling in various subjects, mostly physics and mathematics but also chemistry, biology, and computers. They are always very patient with my foolish...

I am a bit confused about dual space unit vectors in the case of non-orthonormal vector spaces. (Reference: A Student's Guide to Vectors and Tensors, by Daniel Fleisch, Cambridge, 2012.) I will be grateful for being corrected in the following: Suppose V is a non-orthonormal vector space, and...

One of the possible false assumptions here is that mathematics is uniform enough to have the same answer for all of it. It is possible that parts of mathematics are invented, and parts are discovered. The people claiming one side often latch on to that part of mathematics that fits their...

Huge thanks for the replies, Gavran. (The delay in my response was due to my not getting the usual email notification of a reply.) Your examples and explanations are excellent, and well answer my question. Super! Also thanks to you, Wrobel, for the warning, and for your interesting example.

Thanks for the replies, Gavran and Wrobel. First, Gavran. That is good to hear. Could you give me an example of each, and why? (That is, one in which it is better to use an inner product, on in which it is better to use a dual space operation.) That would be super. Second, Wrobel. I am more...

Thanks for answering, Hill, although it appears that either I did not state the question clearly enough, or you read the question rather hastily, as I mentioned neither which inner product was being referred to nor over what field the vector fields were; I neither mentioned nor meant the dot...

Starting with a vector space V equipped with an inner product (. , .), and its dual space V*, one can look at the expression <a|b> in one of two ways It is the dot product ( |a> ,|b> ), with |a> and |b> from V It is the functional <a| from V* applied to |b> from V. Since the two equal the...

Thanks very much, Gavron! I see my error now. I had not rid myself of my prejudice that isomorphism preserved all structures; I understand now that isomorphism only preserves the specific operations that define the structure in question, but that "transpose" is an additional structure...

Thanks, Gavran, but apparently I did not express myself very well. Let me try again. First, if I understand your post correctly (but probably I am not), it seems that in Post #8, that you are stating Then, however, suppose we take the complex conjugate of both sides , then we get where α† is...

So, for example, if we take the Hermitian adjoint (the composition of the complex conjugate and the transpose), I would deduce from your sentence that this would mean that the conjugation of the conjugate of scalar s corresponds to the Hermitian adjoint of its matrix representation; that is, s...

Thanks, Gavron. These simple examples are very nice.