Linear Algebra and Multidimensional Geometry
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W is just the set of things equivalent to zero. I said somehing crazy: "If you pick any line L through the origin and not equal to W, then L meets exactly one line which is parallel to W. Then K is a subgroup of G and we can define a group structure on the set of equivalence classes under the relation, that x and y in G are equivalent mod K if x-y is in K, i.
In fact this is possible if and only if K is the kernel of some group map, if and only if for every element x of G the set of left products xK equals the set of right products Kx, assuming the operation on G is multiplication , if and only if K is "normal". In particular this always holds if G is abelian, as in the case of a vector space where the operation is vector addition.
This is wrong: "Let R denote our equiv. W consists precisely of the elements x which are equivlent to zero, i. Frankly i do not think of an equivalence relation as a set of ordered pairs. Last edited by a moderator: Jan 5, The important thing is that the set of equivalences classes carries a vector space structure. This may seem odd to you, but this kind of thing is exactly where equivalence classes came from, or at least it appears to be: "the" example of them is modulo arithmetic if you like. I've seen very few "good" mathematicians who would choose define equivalence classes as ordered pairs, as subsets of AxA.
This treatment, whilst abstract and general disguises the naturalness of the definition of an equivalence relation and only seems to confuse the issue. W isn't an equivalence relation. We are not playing at all loose with the definitions - you must realize that these are just notational conventions. Thanks for the reply Matt. I've been camping in set theory stuff too much lately. I guess I never thought of "equivalence" for vector spaces in set theory terminology like "equivalence relation", "equivalence classes", "quotent sets", and all that.
But, I'm just a persistent novice and undoubtedly have quite a few misconceptions and distortions in my understanding. I'm working on it. And yeah - I do understand the congruence relation in modulo arithmetic doesn't have anything to do with ordered pairs or cross products Later, Perion. The ordered pair stuff is there to give a sound basis for the theory, just like a function from X to Y is "actually" a subset of XxY.
BUT When it comes to understanding and thinking of equivlance relations on a set A, it's best to think of it as some way of partitioning A into disjoint subsets in some nice manner, working with the equivalence classes as subsets of A and things like that, just as if we were asked to show that f: X to Y is surjective, say, we'd prove something using f, we would never bother to write things out with elements of XxY.
So [x] is the collection of elements in A that are related to x.
In this case [x] means all the vectors in V that differ from x by an element of W, or better y is in [x] if x-y is in W. I have just read sharipov's book and find it pretty nice. I have covered much of his chapter one, but he gives a lot of elementary background on sets and functions which could be useful to beginners. He also gives more details of course than I do. Then I treat more lightly his chapter 2. Then I give a more elementary version of his chapter 3 which leads not to his more general jordan form, but to a result somewhere between it and his spectral theorem as given by him in chapter 5.
Now it turns out the nilpotent part is zero provided the minimal poynomial factors into distinct linear factors. So I only consider the second case and I thus describe the special case of his result where there is no nilpotent part to the operator. Then I give a criterion for this to happen, namely where the matrix is symmetric, what he calls the spectral theorem for self adjoint operators. It is a good book though, and I will probably give it out to my students just in case any of them can read it.
I guess if i had more typographical capacity I could write it as a little equivalence twiddle as you do, with a W at the bottom, to show it emans equivalence relation defiend by W, and not W itself, but we seldom belabor notation like that. Last edited: Jan 5, Thus sending a polynomial P to multiplication by P, defines a linear map from R[X] to the space of linear maps on V. Since the space of n by n matrices is itself a vector space of dimension n2, so also the space of linear maps from V to V has dimension n2, and since R[X] is not finite dimensional, having as basis all monomials 1,X,X2,X3, The monic polynomial of least degree in that kernel is called the minimal polynomial of f.
By the division algorithm, any other polynomial in that kernel is a multiple of the minimal one. Then vt is in the span of the vectors v1, Then vt is also sent to zero by this polynomial, but this is false. Then X-c2 sends ct-c1 vt to ct-c1 ct-c2 vt which is not zero, For surjectivity, let v be any vector in V, and define the polynomials P1, Proof: For a diagonal matrix with entries c1, In fact this is still true when we omit repeated occurrences among the scalars ci.
On the other hand no proper factor factor of this polynomial can annihilate all vectors in V as we saw in the last proof. Thus this is the minimal polynomial. Conversely if the map f has such a minimal polynomial, then V is isomorphic as above to the product of its subspaces ker f-ciId. Choosing bases of each of these subspaces, and taking their union then gives a basis for V consisting of eigenvectors for f.
Perion said:. Here is a site where a short, 15 page, primer of linear algebra can be downloaded free.
It covers dimension, bases, quotient spaces, product spaces, matrices, linear maps, eigenvalues and eigenvectors, spectral theorem for symmetric matrices, relation between minimal polynomials and diagonalizability, characteristic polynomials, jordan form, rational canonical form, and Cayley Hamilton theorem. To get it down to 15 pages, the philosophy is to include or sketch all clever proofs, but leave as exercises all straightforward, or tautological arguments.
Thus it is appropriate for very motivated and talented young students, or as review for those who already know the material, like grad students preparing for prelims. Still some results are proved more than once, first in easier cases then again in more general ones.
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