Post B5BRV4K3orZPI0Q1om by mildsunrise@tech.lgbt
 (DIR) More posts by mildsunrise@tech.lgbt
 (DIR) Post #B5BRV4K3orZPI0Q1om by mildsunrise@tech.lgbt
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       i love my girlfriends so much... i feel so fortunate
       
 (DIR) Post #B5BSgoWCVLsphdGHUe by mildsunrise@tech.lgbt
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       also they are so hot goshhow am I supposed to prove that adj(X) * X = det(X) * Id in this state :blobcatnotlikethis:
       
 (DIR) Post #B5BZpyC0bXURzb87e4 by raito@nixos.paris
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       @mildsunrise i'd just diagonalize X and call it a day
       
 (DIR) Post #B5BZpyO3sj8CaylkQq by raito@nixos.paris
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       @mildsunrise some density arguments about matrices which eigenvalues are distinct and nonzero because laziness
       
 (DIR) Post #B5BZpyZlBEUNBGF5fM by mildsunrise@tech.lgbt
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       @raito uhhh that sounds like it demands more than what i have, nobody said X has to be diagonalizable or invertible and also the underlying structure is just a (commutative) ring
       
 (DIR) Post #B5BZpyjgaKQdg2t18a by raito@nixos.paris
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       @mildsunrise doesn't matter, the set of diagonalizable matrices with distinct and nonzero eigenvalues is dense, so you can just do the work with X diagonalizable with distinct and nonzero eigenvalues (easy mode), then you take a sequence and you use continuity of the different functions to obtain the same result
       
 (DIR) Post #B5BZpyufvTDeE81nGa by mildsunrise@tech.lgbt
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       @raito i don't have topology developed yet (and I barely know any myself) and the underlying structure is just a bare ring, I'm not demanding it to be a topological space... also even if I did, are you sure "the subset of diagonalizable matrices is dense" doesn't require certain things on the underlying topology? for example I think the matrices over a Galois field have a discrete topology, so the only dense subset would be the entire space i believe?
       
 (DIR) Post #B5BZpz5JHvj4l70HqK by raito@nixos.paris
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       @mildsunrise it requires reasonable thing on the topology, if you do zariski topology, it might be cooked, i'm not sureif you don't have topology, i guess you need to do chain rules related stuff or calculate the differential of det at the identity matrix point (which also makes use of chain rule somewhat), it feels weird to me because all of these requires some sort of topology, i don't think there's a real pure algebraic solution here?