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Thank you for your answer. I've already read the wikipedia article, but still fail to understand what we mean by "vector space". Google says it is "a space consisting of vectors", but the first example on the wiki page (arrows in the plane) confuses me:
- is a vector space the parallelogram formed by the vectors?
- is a vector space the ensemble of all vectors existing within that plane?
- is a vector space something else altogether?
And then, once I will have understood what it is, to prove that V is a vector space, do I have to check all 8 of the axioms and see if they are valid?
Thank you very much for your help, I appreciate it.
J.
Others can probably give you more detailed information but after reading the Wikipedia article I think your confusion centers around your concept of a vector in mathematics vs physics.
On one hand we think of a vector as that arrow on a plane or in 3-D, and we think of the "space" as the plane or the volume within which the arrow is placed. And, indeed, as the article points out that is where the primitive notion arises. Later, for the purpose of developing an algebra of vectors, that notion is replaced by the concept of a vector as an n-tuplet, namely an ordered pair (x, y) on a plane (R
2) or an ordered triplet (x, y, z) in R
3. With the concept of a vector as an n-tuplet one can now conceive of a vector in some abstract space R
n.
Early on we are taught how to add/subtract, multiply/divide numbers, 2 + 3 = 5, 5 x 6 =30 etc. The question now is how to add pairs or triplets of numbers, what do you do if given (1,5) + (2,3), or (1, 6, 7) + (7, 8, 2). You need rules defining what those expressions mean and are equivalent too.
A large part of mathematics is formalized by abstracting the essence of something and generalizing on it. The expressions 2 + 3, and (1,5) + (2,3), can be both be generalized as, in essence, "expressions" of an operation on a pair of objects. In the first case the objects are numbers, in the second case, n-tuplets (specifically an ordered pair). In both cases the operation is one of "addition", but, because the objects being added are different in each case the rules/procedure for adding the objects are different.
Now, depending on the objects under discussion, numbers, vectors, matrices, etc, different operations may or may not be relevant. For example, if one is analyzing chess moves where the the board and chess pieces comprise a set of objects, do the operations of addition and multiplication apply? Probably not, you would need to define a completely different set of operations to define an "algebra of chess".
The main point is that while one might find it useful to think of vectors as arrows in space when representing a physical conception, mathematics is more abstract. In mathematics, I would suggest, a "vector" is not an arrow but an ordered n-tuple. Correspondingly, a vector space is not a geometric plane, or a volume, but rather a set or subset of ordered n-tuplets over which the defining properties of a vector hold ... and that is what this exercise is establishing. In general, unless you are going to be a mathematician proving basic properties is just done once, then you just use them.
As to the proving part, this might be an example:
Let A be the vector (a,b), let B be the vector (c,d) where a,b,c,d are real numbers.
Is it true that the addition of vectors is commutative? Does A + B = B +A
By the definition of the addition of ordered pairs:
A + B = (a,b) + (c, d) = (a+c, b+d)
B + A = (c, d) + (a, b) = (c + a, d + a)
but real numbers are commutative over addition so:
(c + a, d + a) = (a + c, b + d), therefore
A + B = B + A and so vectors are commutative in 2-D, does one need to extend this to n-tuplets? Don't know, but that is the idea.
.
