importance of Krylov subspace Can someone please explain to me why we are using Krylov subspaces for the CG-Method, GMRES-Method and Arnoldi.
Unfortunately I do not see where the advantages are und why Krylov subspaces are so important/special.
In the literature, it says that the Krylov subspace are search spaces, but I don't understand why
I hope someone can help me
 A: In optimisation you normally work in a particular space, known as your search space. An example of a simple search space is $\mathbb{R}$ which are the real numbers. Another example is $\mathbb{R}_{>0}$. That is, positive real numbers. 
The main aim of conjugate gradient methods is to solve the following equation: $Ax = b$ where $x$ is a vector of unknown quantities, $A$ is symmetric positive definite, and $b$ is known also. It can be difficult to find the inverse for large, sparse systems, so that $A^{-1}$ is hard to calculate. Instead we can use optimisation to find the optimal $A$ to solve this system of equations.
So we have an optimisation problem (find the optimal $A$) and we know optimisation problems need a search space. But the search space is a bit weird now, since we search over an entire matrix! Not something easy like the reals. More accurately, we would like to minimise: $r := b - A x \,$ the residual error.
Now the Kyrlov subspace is defined as follows:
\begin{equation}
{\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.
\end{equation}
In the conjugate gradient optimisation scheme there are lines in the algorithm that look similar to things like: $A^{r-1}b$.
Therefore we just say the solution of the conjugate gradient involves spanning the Kyrlov subspace. It is more of a technical note, since remember optimisation needs (i) an objective, (ii) a space on which the solution must exist. 
It is more than a technical note, if you want to do research by which then it is important to know the properties of the Kyrlov subspace! It could help in modelling constraints, understanding convergence rates, understanding convergence behaviour etc...
It is similar to understanding Galerkin methods for Finite Element Analysis.... Sure theoretically you need to know Galerkin theory for FEA, but if you are a practising engineer, this knowledge is just a complication. BUT, if you are a researcher, it would be important to go through the derivation, and understand the inner workings of weak formulations, the functions they can permit etc...
This answer goes more into the mathematics intuition of what I just discussed:
https://math.stackexchange.com/a/2714985/580635
