By Boij M., Laksov D.

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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. We say that f is differentiable in U if it is differentiable at each point of U . 12. Usually the linear map f (x) is represented by an m×n matrix with respect to the standard bases of Kn and Km and the distinction between the matrix and the map is often suppressed in the notation.

1. We have that Ψ (Wi ) ⊆ Wi , for i = 1, 2. Indeed, write si = sei for i = 1, 2. We 1 ),e2 e2 ) = Ψ (e1 ) − have that −Ψ (e1 ) = Ψ (s1 s2 (e1 )) = s1 s2 (Ψ (e1 )) = s1 (Ψ (e1 ) − 2 Ψ (e e2 ,e2 1 ),e1 1 ),e2 1 ),e1 1 ),e2 2 Ψ (e e1 − 2 Ψ (e e2 . Consequently, Ψ (e1 ) = Ψ (e e1 − Ψ (e e2 . Similarly it e1 ,e1 e2 ,e2 e1 ,e1 e2 ,e2 follows that Ψ (e2 ) ∈ W2 . A similar argument, with indices n, 1 instead of 1, 2 gives that Ψ (W1 ) ⊆ W1 . We obtain that Ψ (W1 ∩ W2 ) ⊆ W1 ∩ W2 . Consequently we have that Ψ (x) = ax, for some a ∈ K.

Let U be the ball B(In , 1) in Gln (K) and let V = log(U ). The following five properties hold: (i) log exp X = X, for all X ∈ Mn (K) such that log exp X is defined. (ii) exp log A = A, for all A ∈ Gln (K) such that log A is defined. (iii) det exp X = exp tr X, for all X ∈ Mn (K), where tr(aij ) = ni=1 aii . (iv) The exponential map exp : Mn (K) → Gln (K) induces a homeomorphism V → U . The inverse map is log |U . (v) log(AB) = log A + log B, for all matrices A and B in U such that AB ∈ U , and such that AB = BA.