If $A$ be an $n$ by $n$ matrix. Prove that $$\dim(\textrm{span}({I_n,A,A^2,....}))\leq n$$
I don't know, how can i prove it. Please help me. Thanks
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Maybe this helps – GAVD Oct 09 '15 at 03:54
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@GAVD That looks like it might be more relevant than the question currently being suggested as the duplicate. – Robert Soupe Oct 09 '15 at 18:01
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This is a direct consequence of Cayley-Hamilton theorem, which implies that $A^n$ is a linear combination of $I,A,A^2,\ldots,A^{n-1}$ and hence higher powers of $A$ are also linear combinations of $I,A,A^2,\ldots,A^{n-1}$.
If you want to avoid using Cayley-Hamilton theorem, you may see "Degree of minimum polynomial at most n without Cayley-Hamilton?" and its related postings.
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As user1551 mentions, characteristic polynomial together with Cayley-Hamilton theorem will do the trick, but actually minimal polynomial will suffice, and it can be of lower degree.
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