LADR Finding Eigenvectors to Find a Dependence

Linear Algebra Done Right Theorem

1. Suppose $V$ is a $n-dim$ real vector space
2. Then. if $T:V\to V$ is a Linear Map
3. Then, $\{I,T^1,\dots ,T^{n+n}\}$ is linearly dependent in $L(V,V)$
4. Alternatively, there are non-zero constants such that $Z=a_{n+n}{T}^{n+n}+\cdots +a_1{T}^1+a_{0}I$
5. Where $T:V\to V$ is the identity and $Z:V\to V$ is the zero map

Examples

Find a linear dependence among the powers of

$$\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$$

Soln

$$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$ $$T^1=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$$ $$T^2={\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]}^2=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

This gives $T^2=T+I$ Thus, $T^2-T-I=Z$ Thus, our set $\{I,T,\dots ,T_n\}$ is dep