Define a Krylov subspace $\mathcal{K}_{n}(A, v)$.

Let $d_{n}$ be the dimension of $\mathcal{K}_{n}(A, v)$. Prove that the sequence $\left\{d_{m}\right\}_{m=1,2, . .}$ increases monotonically. Show that, moreover, there exists an integer $k$ with the following property: $d_{m}=m$ for $m=1,2, \ldots, k$, while $d_{m}=k$ for $m \geqslant k$. Assuming that $A$ has a full set of eigenvectors, show that $k$ is equal to the number of eigenvectors of $A$ required to represent the vector $v$.