$T:V\to W$, both has same basis Suppose $W,V$ have the same basis $\{u_1,u_2\}$ and that $T:V \to W$ is a linear transformation. Give an example (not the identity transformation) of 
a) $T = T^{-1}$
b) $T= T^2$
for a) $T=T^{-1} \Rightarrow A=A^{-1}$
let $A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right]$
$A=A^{-1} \Rightarrow A^2=I$ $A^2 =  \left[\begin{array}{cc}a^2+bc&ba+bd\\ca+cd&d^2+bc\end{array}\right] =  \left[\begin{array}{cc}1&0\\0&1\end{array}\right]$
$a^2+bc-bc-d^2=1-1 = a^2 - d^2 = (a-d)(a+d)=0\\c(a+d)=0\\b(a+d)=0$
From here though I'm a tad iffy, like $ \left[\begin{array}{cc}2&3\\0&-2\end{array}\right]$ would satisfy what I got solving the system but $A \neq A^{-1}$ but if I do $ \left[\begin{array}{cc}1&3\\0&-1\end{array}\right]$
$A =A^{-1}$ and $A^2=I$
I'll do (b) after I get this one.
thanks!
 A: It is good practice to work through these exercise using the nitty-gritty algebra, but these questions are also a good opportunity to develop geometric intuition.
In particular, you've probably seen that three important geometric operations are linear transformations:


*

*Rotations about the origin $T=\left[\begin{array}{cc} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{array}\right]$;

*Projection onto the line spanned by the vector $v$: $T=\frac{vv^T}{\|v\|^2}$;

*Reflection about the line perpendicular to the vector $n$: $I - 2\frac{nn^T}{\|n\|^2}$.


Now part a) is asking for a transform that is the same as its inverse, i.e. when applied twice gives you the identity. And part b) is asking for a transformation that does the same thing when applied twice as when applied once, in other words, after applying it once, applying it a second time doesn't do anything additional. Examples of both of these can be readily found by thinking about the above geometric transformations.
