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Two Linear Algebra exercises

Linear Algebra
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Let be \[A,B \in K^{n x n}\] such that A.B=B.A and let \[E_{\lambda} = \left\{x \in K^{n} / A.x = \lambda . x \right\}\] Prove that \[E_\] is invariant
\[E_{\lambda}\] *
Lets be \[A,B \in K^{n x n}\] two diagonalizables matrix such that A.B = B.A Prove that there's \[C \in GL(n,K)\] which \[C.A.C^{-1}\] and \[C.B.C^{-1}\] are diagonals. (With the same C).

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Other answers:

K is a generic field.
I guess I'll have to keep trying xD
you should try Linear algebra by david c lay 4th edition..chapter 5 eigen values and eigen vectors..i can give you if you want
I already read some theory, but reading more never hurts, so feel free to give me it to me :D
lol ok wait

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