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- Mar 10, 2012

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said to have rank $r > 0$ w.r.t eigenvalue $0$ if $T^rv=0$ but $T^{r-1}v\neq 0$. Let $x,y \in V$ be linearly independent and have

ranks $r_1$ and $r_2$ w.r.t eigenvalue $0$ respectively. Show that $\{x,Tx,\ldots ,T^{r_1-1}x,y,Ty,\ldots , T^{r_2-1}y \}$ is a

linearly independent set of vectors.

I can see that $\{ x, Tx, \ldots , T^{r_1-1}x \}$ are Linearly Independent, and $\{ y, Ty, \ldots, T^{r_2-1}y \}$ are linearly independent, but now I am stuck. Please help.