Understanding Vector SpacesDate: 03/14/2001 at 09:31:47 From: Matea Subject: Understanding vector spaces I have been trying to read about vector spaces everywhere, and I just can't seem to grasp the concept. Can you help me? I don't know how to think about them. What does a vector space in R^n mean? How can I prove that a list of numbers is a vector space? Thanks! Matea Date: 03/14/2001 at 10:30:26 From: Doctor Mitteldorf Subject: Re: Understanding vector spaces Dear Matea, I can remember having the same question as a college student. For me the problem turned out to be that I had all kinds of associations with the words "vector" and "space," and I couldn't attach them to the examples that were being given. The key was just to think of "vector space" as just some nonsense syllables that described a collection of objects obeying a few simple postulates: 1. There must be a rule for "addition" of two objects: The space must be closed under addition: if X and Y are in the space, then X+Y should be in the space as well. Addition must commute: X+Y = Y+X. 2. There must be a rule for "multiplication" of any object by a number: The space must be closed under scalar multiplication: if X is in the space, and a is a number, then aX should be in the space as well. 3. One of the objects in the space must serve as a "zero." The scalar number 0 multiplied by any member of the space gives this zero vector, and the zero vector can be added to any other vector X without changing it. 4. Addition in the space must parallel addition of scalars. In algebraic language, this can be stated: aX + bX = (a+b)X If, like me, you're accustomed to thinking of vectors as arrows in our 3D space that represent force or displacement, then you can see that these postulates are true of force vectors. But they're also true for collections of objects that might seem less "physical": Colors made up from three primary colors form a 3-D vector space. Profiles of test scores on a battery of n tests can be seen as an n-dimensional vector space. There is a vector space of all infinite sequences, and a vector space of all real-valued functions on a given domain. These spaces obey the postulates, even if they don't seem to have much to do with arrows or forces. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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