User talk:Markus Schmaus

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Various questions

Also a warm welcome from me. I saw that you have been contributing to some articles on mathematics, so I thought you might be interested in Wikipedia:WikiProject Mathematics which discusses Wikipedia issues specific to maths.

I do have two remarks about your edits:

There seems to be a mistake in the article Family (mathematics), as I stated on Talk:Family (mathematics).
I reverted your edit at Linear independence. It is true that a projective space is not a vector space, but when $(a_{1},...,a_{n})$ is introduced, it is not yet seen as an element of the projective space. This only happens in the third sequence ("It makes sense to ...").

Anyway, I look forward to your future contributions. Feel free to add me any questions on my talk page, which is at User talk:Jitse Niesen. Tschüss, Jitse Niesen 04:29, 15 Jun 2005 (UTC)

Did you want to write something on my talk page? It now has an heading Families, but no text. -- Jitse Niesen 14:34, 16 Jun 2005 (UTC)

Yes, I present enter after starting with the heading, which triggered save page. I later was interupted writing my message.

Families

Hi, we have some discussion going on, all of them in some way connected to families. Maybe it's a cultural thing. My German professor defined a basis as a family and I found lots of German sources doing the same. But I actually found hardly any English sources doing this. Generally families don't seem too popular in the English math community. Mathworld uses the curly brackets notation for families, which I find misleading for the reasons stated in my article.

Families is a formalisation of statements as "let v₁, ..., v_n be a basis" or "[a matrix A is invertible, if] the columns of A are linearly independent" (taken from invertible matrix).

It is perfectly possible to do linear algebra without using index notation, I actually prefer index free notation

\int _{\Omega }f

over

\int _{x\in \Omega }f(x).

I like universal properties and there is no need for families when talking about a free module over a set, the corresponding basis.

But many people, especially physicists, prefer index notation. They like to talk about first, second, or third coordinates, they use matrices instead of homomorphisms. Index notation can be formulated exactly, but this requires using families.

Making index free definitions and having pages using index notation refering to those definitions leads to small inconsistencies as is the case with invertible matrix. Markus Schmaus 15:42, 16 Jun 2005 (UTC)

English mathematical texts are indeed generally less formal than text from continental Europe in that they tend to put more emphasis on readability, sometimes at the expense of rigour; at least, that is my impression. And then there is the difference between applied and pure maths, and maths generally and physics. However, I don't actually see the small inconsistency in invertible matrix that you are referring to. Is it in the phrase "the columns of A are linearly independent"? But the "columns of A" can just as well refer to the set of columns as to the family of columns. I agree that "let v₁, ..., v_n be a basis" is not precise; it should properly be written as "let {v₁, ..., v_n} be a basis".

I hope you understand that I am not following you and trying to obstruct all you do, but I was rather surprised when somebody starts using unfamiliar terminology in some of the articles on my watchlist. -- Jitse Niesen 17:10, 16 Jun 2005 (UTC)

Consider the matrix ( (1,1)^T, (1,1)^T ), it is not invertible, but the set of columns is lineraly independent, the family of columns wouldn't be. Markus Schmaus 17:22, 16 Jun 2005 (UTC)

You are absolutely right. Let me consider what the best way is to resolve this. I am still hesitant to define basis as a family, since many will not be familiar with that term. Perhaps the standard English term is sequence. However, there is a difference between family and sequence, in that a sequence is linearly ordered. -- Jitse Niesen 17:51, 16 Jun 2005 (UTC)

A sequence is a family with the natural numbers as the index set. In the old version of index (mathematics) the term indexed set was used synonymously to family. But I have never heard that term before and I didn't find many references on google. Array might be an idea, but I don't think array is a common and well defined mathematical term. I still think using family would be best, but I will try to make family (mathematics) easier to understand.