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Let $V$ be a vector space over $\mathbb C$ with $\dim V=n$ and $F\colon V\to V$ be a linear map.

(a) Show that there always exists a basis $\{v_1,\ldots,v_n\}$ such that $F(v_j)$ is in space $W_j$ which is generated by $\{v_1,...,v_j\}$

(b) Is part (a) true if we consider a linear map $F\colon V\to V$ where $V$ is a vector space over $\mathbb R$? Justify your answer by giving a counterexample or a proof.

I want to solve this question T T

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    Are you familiar with A) eigenvectors, B) Jordan canonical form, C) rotations?2012-04-28
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    Use induction. See for instance [Schur decomposition](http://en.wikipedia.org/wiki/Schur_decomposition).2012-04-28

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