Let $V$ be a vector space and $T : V\to V$ a linear transformation with the property that $T(W)\subset W$ for every subspace $W$ of $V$. How can we prove that there is an element $\lambda$ in the field of scalars such that $T(v) = \lambda v$ for all $ v \in V .$
eigenvector proof
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linear-algebra
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4And [another](http://math.stackexchange.com/q/116223/742) strange bounty offer. What is "non-credible" about the answers given? That they didn't spoon-feed you the full answer in complete detail? If you want more details, ask for details. Saying the answers you received are "not credible" is rather insulting. As for "official sources", sorry, I won't fax you a copy of my degree, but it's pretty official. It says I have "all the rights and priviledges thereto pertaining", so I'd say I'm an official source. – 2012-03-23