If I have an $n$-by-$n$ matrix $A$, is $1, A, A^2,...,A^{n^2}$ always linearly independent?
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No. For instance, you can have $A=0$ (the zero matrix), or $A=I$ (the identity). I guess you can ask when they are linearly independent, but I'm not sure if this is known – Mankind Jun 02 '15 at 23:08
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4Cayley-Hamilton says $A$ satisfies a polynomial of degree...? – David Wheeler Jun 02 '15 at 23:08
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No, never, because the space of $n\times n$ matrices has dimension $n^2$ and you have $n^2+1$ matrices there.
lhf
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Ugh. Sorry. That was dumb. Would the first 1, A, ..., A^{n^2-1} matrices be linearly independent? – onemillion6 Jun 03 '15 at 00:03
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