How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular. $\mathbf{K}$ is a Krylov matrix.
\begin{align}
 \mathbf{K}&= \left[ \begin{array}{ccccc}
  \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots &  \mathbf{A}^{N-1}\mathbf{b} \end{array} \right]^T.
\end{align}
How to prove that the vectors of the Krylov space of $\bf A$ are linearly independent if $\bf A$ is nonsingular?
 A: In general this is not true, for example ${\bf A} = k{\bf I}$ or a rotation matrix in 3D where vector projected along the the axis of rotation will never change and therefore always be linearly dependent. More generally any linear operator projecting on a specific non-zero dimensional subspace and having any arbitrary non-singular transform on the rest of the space, the vectors mentioned will not be linearly independent.
These are usually the cases in which Krylov subspace methods will converge faster.

But for example if all eigenvalues have distinct modulus, then the eigenspace must be simple and the matrix must be diagonalizable. Then each eigenvector (let us call them $\bf v$) will be scaled so we can write $${\bf A}^n{\bf b} = \sum_{\forall k} {\lambda_k}^n (\xi_k{\bf v}_k)$$
Since the ${\bf v}_k$ by definition must be linearly independent, the $\lambda_k$ are distinct and the monomial functions are linearly independent, we are done. This usually means we need to investigate larger parts of the space (more iterations) which makes convergence become slower.

Another way to maybe easier realize this is to write the coefficients of the vectors into a matrix:
$$\left[\begin{array}{lllll}\xi_1&\xi_2&\xi_3&\xi_4&\cdots\\\xi_1\lambda_1&\xi_2\lambda_2&\xi_3\lambda_3&\xi_4\lambda_4&\cdots\\\xi_1{\lambda_1}^2&\xi_2{\lambda_2}^2&\xi_3{\lambda_3}^2&\xi_4{\lambda_4}^2&\cdots\\\xi_1{\lambda_1}^3&\xi_2{\lambda_2}^3&\xi_3{\lambda_3}^3&\xi_4{\lambda_4}^3&\cdots\\\xi_1{\lambda_1}^4&\xi_2{\lambda_2}^4&\xi_3{\lambda_3}^4&\xi_4{\lambda_4}^4&\cdots\end{array}\right]$$
If these $\lambda_k$ are distinct (and our $\xi_k \neq 0$), what happens if we try to row-reduce?
