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MathematicalPhysicist

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now for the first group I tried to do it with brute force, but with no success, i.e SO(n) are the nxn orthogonal matrices with determinant 1, so we need to find a function f:[0,1]->SO(n) s.t f(1)=A f(0)=B for every A,B in SO(n), so as always i tried to use this function: f(x)=xA+(1-x)B where x in [0,1] but I need to show that the multiplication of it with its transpose gives the identity matrix, any ideas here?

also if you can help me with the other groups it would help very much.

thanks in advance.