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

By Boij M., Laksov D.

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Are Cauchy in K, for k = 1, . . , n. Since the real and complex numbers are complete we have that these sequences converge to elements a1 , . . , an . It is clear that x1 , x2 , . . converges to x = (a1 , . . , an ). 3. For X in Mn (K) and m = 0, 1, . . , let expm (X) be the matrix 1 1 1 m expm (X) = In + X + X 2 + · · · + X . 1! 2! m! The sequence {expm (X)}m=0,1,... is a Cauchy sequence in Mn (K) because, for q > p, we have that 1 1 X p+1 + · · · + X q (p + 1)! q! 1 1 1 ≤ X p+1 + · · · + Xq ≤ X (p + 1)!

N. 2 i∈J Then, for h ∈ P (0, r ), we have that |d1,j ||h|j , . . , |ci |( i∈I j∈J |dn,j ||h|j )i ≤ j∈J |ci | i∈I s 2 i < ∞. Consequently, we have that d1,j y j , . . 1 represents f g(y). 11. Let U be an open subset of Kn and let f : U → Km be a function. If there exists a linear map g : Kn → Km such that lim h →0 f (x + h) − f (x) − g(h) = 0, h where h = maxi |hi |, we say that f is differentiable at x. Clearly, g is unique if it exists, and we write f (x) = g and f (x)h = g(h), and call f (x) the derivative of f at x.

It is clear that Ψ is a linear map. The linear maps of this form are called transvections. 6. We have that each transvection is in Sp(V ). 6 we can choose a symplectic basis e1 , . . , en for the bilinear form with x = e1 . Then we have that Ψ (ei ) = ei for i = n and Ψ (en ) = en + ae1 . Hence for y = ni=1 ai ei we obtain that Ψ (x), Ψ (y) = e1 , ni=1 ai ei + ae1 = e1 , ni=1 ai ei = x, y . We also see that det Ψ = 1. 7. Let , be a non-degenerate alternating form on V . Then for every pair x, y of non-zero vectors of V there is a product of at most 2 transvections that sends x to y.