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

1. let V be vector space. <br />
\dim V = n,n > 1<br /> <br />
and let T:V-V be a linear transformation that sustains dimImT=1 and <br />
{\mathop{\rm Im}\nolimits} T \not\subset \ker T<br /> <br />
.
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 <br />
\lambda  \ne 0<br /> <br />
such as <br />
\left[ T \right]_B  = \left[ {\begin{array}{*{20}c}<br />
   0 & 0 & 0  \\<br />
   0 & 0 & 0  \\<br />
   0 & 0 & \lambda   \\<br />
\end{array}} \right]<br /> <br />


2. given unhomogane set of linear equations in 4 veriables, that the vectors <br />
(0,1,2,3),(1,2,3,4),(1,2,4,5),(5,8,3,1)<br /> <br />
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 <br />
A^2,B^2 <br /> <br />
are also similar!!!

Thanks ahead (srry about the bad english)