Consider a linear map $T:V \to V$. Choose one non-zero element (suppose it exists) and take its span. Suppose this subspace is invariant under $T$. Does that imply that any complement of this one-dimensional subspace will also be invariant under $T$. I don't think so.
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http://math.stackexchange.com/questions/258502/do-t-invariant-subspaces-necessarily-have-a-t-invariant-complement – Prahlad Vaidyanathan Sep 05 '13 at 13:52
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@PrahladVaidyanathan : But suppose I consider the quotient space with respect to that one-dimensional space. Let $v,w \in V$ belongs to the same element in the quotient space then $T(v),T(w)$ also belongs to the same element in the quotient space. So, we get some kind of invariance. Is not this contradictory (because quotient space is isomorphic to complement) ? Am I making sense ? – aaaaaa Sep 05 '13 at 14:10
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Take $$ T=\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right), $$ the vector $\left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right)$ and a complement Span$\left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right).$
Boris Novikov
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