Geometry And Topology

An introduction to algebra and geometry via matrix groups by Boij M., Laksov D.

By Boij M., Laksov D.

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Show that there are reclections that are not of the form sx for any x in V . 3. Show that Sl2 (K) = Sp2 (K), and write all elements in these groups as products of transvections. 1. Let G be a group. The set Z(G) of elements of G that commute with all elements of G, that is Z(G) = {a ∈ G : ab = ba, for all b ∈ G} is called the center of G. It is clear that Z(G) is a normal subgroup of G and that isomorphic groups have isomorphic centers. 2. The center of Gln (K) consists of all scalar matrices, that is all matrices of the form aIn for some non-zero element a of K.

6. Assume that the form is alternating. We then have that n = 2m is even and there is a basis e1 , . . 8) is of the form S= 0 Jm , −Jm 0 where Jm be the matrix in Mm (K) with 1 on the antidiagonal, that is the elements aij with i + j = m + 1 are 1, and the remaining coordinates 0. Moreover the basis can be chosen so that it contains any given non-zero vector x. Proof: If n = 1 there is no non-degenerate alternating form. So assume that n > 1. Let e1 be an arbitrary non-zero vector. Since the form is non-degenerate there is a vector v such that e1 , v = 0.

The first term is zero if and only if x = x,y , hence equality holds if and only if x = ay for some y 2 a. 4. Let (X, d) be a metric space. Show that the collection U = {Ui }i∈I of open sets satisfies the following three properties: (a) The empty set and X are in U. (b) If {Uj }j∈J is a collection of sets from U, then the union ∪j∈J Uj is a set in U. (c) If {Uj }j∈K is a finite collection of sets from U, then the intersection ∩j∈K Uj is a set in U. 5. Show that if f and g are continuous functions Kn → K, then cg, f + g, and f g are continuous for each c in K.

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