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.
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