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I came across the following problem that says:

Let $C$ be a $n\times n$ real matrix. Let $W$ be the vector space spanned by $\{1,C,C^2,\ldots,C^{2n}\}$.

Then, which of the following about the dimension of the vector space W is/are correct?

The dimension of $W$ is:

  1. $2n$
  2. at most $n$
  3. $n^2$
  4. at most $2n$.

Please help.Thanks in advance for your time.

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C must satisfy a polynomial of degree at most n. Thus the dimension is at most n. –  lee Dec 7 '12 at 14:24
    
think about $C$ being the zero matrix to exclude 1 and 3. –  Thomas Dec 7 '12 at 14:28
    
@Thomas It is a multiple choice question ,sir.So more than one option are possible,as is evident from the question.Can u please explain sir,why option (c) is also correct provided (b) is correct? –  learner Dec 7 '12 at 14:44
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1 Answer

up vote 2 down vote accepted

First, if you consider the example where $C$ is the zero matrix, you can eliminate option 1. and option 3.

You are left with the other two options. Note then that if 2. is correct, then obviously 4. is correct as well. (If the dimension is always at most $n$, then since $n\leq 2n$, the dimension is also always at most $2n$.)

It is all just about determining whether 2. holds.

However, if you know about the Minimal polynomial you realize, as @lee mentioned in the comments, that $C$ has to satisfy a polynomial of degree at most $n$ (since $C$ is an $n\times n$ matrix).

So among the set of matrices in $\{1, C, C^2 , \dots , C^{2n}\}$, you can have at most $n$ linearly independent matrices.

And so $\dots$

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thank you sir,for the clarification. –  learner Dec 7 '12 at 14:59
    
@learner: Glad to help. –  Thomas Dec 7 '12 at 15:00
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