I have a test soon I'll apprieciate if you'll help me:

1. let V be vector space. $\displaystyle

\dim V = n,n > 1

$ and let T:V-V be a linear transformation that sustains dimImT=1 and $\displaystyle

{\mathop{\rm Im}\nolimits} T \not\subset \ker T

$.

prove that T has an eigenvalue that is different from zero.

let n=3. prove that V has a basis B and there is a scalar $\displaystyle

\lambda \ne 0

$ such as $\displaystyle

\left[ T \right]_B = \left[ {\begin{array}{*{20}c}

0 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & \lambda \\

\end{array}} \right]

$

2. given unhomogane set of linear equations in 4 veriables, that the vectors $\displaystyle

(0,1,2,3),(1,2,3,4),(1,2,4,5),(5,8,3,1)

$

are solutions to it. find the general solution of the system!!

3. let A,b be nxn matrixes. given that (I-A) and (I+B) are similar prove that $\displaystyle

A^2,B^2

$ are also similar!!!

Thanks ahead (srry about the bad english)