
Re: Failing Linear Algebra:
Posted:
Apr 28, 2004 11:08 PM


On 28 Apr 2004 23:28:17 GMT, Anonymous wrote:
>Grubb: > >>>I don't understand. A vector is any collection: (x1, ...., x^n) of >>anything. >>>In math, the vectors are numbers.
That last sentence is the one he was responding to. The sentence before it is wrong too, but his comments aren't directed to that.
>> >>This is wrong. A function can be a vector (usefully so, in fact). >>A polynomial can be a vector, a sequence can be a vector. In fact, >>*anything* can be a vector. > >Isn't that what I said though?
Sort of, but then why did you say "in math vectors are numbers"?
I said a vector can be a collection of >anything, even if that "collection" just contains one vector: for example, one >function, one polynomial, etc, like you say.
That much is OK, but your notation was wrong. You should use braces {} rather than parentheses when listing the members of a set, and also your notation implies that a vector space can only have a finite number of members; in fact that is only rarely true, and is never true (except for the trivial case {0}) for vector spaces on the fields you're familiar with. E.g. the set of real numbers is a vector space and has infinitely many members. One dimension, yes, but infinitely many members.
Take care that you don't confuse the set of vectors (i.e. the vector space itself) with the ntuple representation of one particular vector in the space according to some basis. The ntuple is what you write with parentheses; it may have a lot of components but it is *one* vector. Btw even here I am not being sufficiently abstract; there exist infinitedimensional vector spaces, and for those spaces the ntuple representation of individual vectors is not possible.
You made the same mistake in your response to me, where you talked about (0,1) as if it were two vectors in R^2, instead of a *single* vector with two components.

