8.40 proof Axler: How to understand

Question

At the start of the proof of minimal polynomial, $T \in \mathcal{L}(V)$ is defined with $dim V=n$. So that the list $I,T,T^2,...,T^{n^2}$ is not linear independent in $\mathcal{L}(V)$, because the vector space $\mathcal{L}(V)$ has dimension $n^2$. I don't really see why the list is not linear independent.

Answer 1

If $\{v_1,\ldots,v_n\}$ is a basis of $V$, define, for each $i,j\in\{1,2,\ldots,n\}$, $E_{ij}\colon V\longrightarrow V$ such that $E_{ij}(e_i)=e_j$ and that $E_{ij}(e_k)=0$ if $k\ne i$. Then $\{E_{ij}\mid i,j\in\{1,2,\ldots,n\}^2\}$ is a basis of $\mathcal L(v)$ and therefore $\dim\mathcal L(V)=n^2$. So, any set of elements of $\mathcal L(v)$ with more than $n^2$ elements is linearly dependent.