Talk:Linear transformation

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Endomorphism f from V into V, where V is a vector space. is it nessary f to be one-one mapping?

Nope.

Part about solving a system of equations ("f(x)=0 is called...") should probably be moved somewhere else. What about other articles about that topic?

[edit] dimension formula only in finite dimensions?

the article states that the formula dim ker f + dim im f = dim V is only valid if V is finite dimensional. Is that really true? Can someone give me a counterexample? -July 2, 2005 01:19 (UTC)

The formula either doesn't make sense (initially) or isn't very useful (if you define addition on infinite cardinals somehow) in the infinite dimensional case. By and large, you're likely to get formulae such as , which are not very useful. If you're really prepared for some heavy mathematics, however, you might want to check out Fredholm operators which have an associated index, a finite number that in the case where V, W are finite dimensional is just dimV - dimU (by the formula in the article). The index has many mystical properties I wot not of. Ben 13:02, 10 August 2006 (UTC)

[edit] Clarification of my move

I moved a pagraph to the bottom of the article, with the edit summary

moved the continuity section to the very bottom. Linear transformations are about vector spaces. For continuity, you need a Banach space. So, continuity is not the primary concern of this article.

Let me clarify myself. First of all, I definitely agree with what the paragraph says, that linear transformations are not necessarily continuous. But the problem is the following. Linear trasformations are about vector spaces. That vector space can be the reals, the complex numbers, a vector space of over a field finite characteristic, over a field which is a Galois extension, etc. All that matters in a vector space is addition and multiplication by scalar.

As such, inserting in the middle of that article a paragraph operators on a Banach space (or if you wish, a linear topological space) was wrong. It distracts the reader from the main point, which is the linearity, addition and multiplication by scalars. To talk about continuity you need topology, you need a norm. It is a totally different realm than the one of a vector space. That's why inserting that continuity paragraph was out of place. It has to of course be mentioned somewhere, but since all the other topics in this article are closely bound together, I put this periferial one at the bottom. Oleg Alexandrov 18:34, 24 September 2005 (UTC)

Ok, that is fine.--Patrick 00:07, 25 September 2005 (UTC)

[edit] Examples

Why are the examples talking about eigenvectors and eigenvalues? Is this not a distraction? --anon

I agree. I cut that out. Oleg Alexandrov (talk) 15:32, 20 July 2006 (UTC)