What makes Krylov subspaces so special?

A good way to motivate the use of Krylov subspace methods is the following idea, which is closely related to a derivation of the conjugate gradient method. $\newcommand{\dotp}[2]{\langle #1,#2 \rangle}$

Assume, we have an initial guess $x_0$ for our solution $x$ of the linear system $$ A x = b \,, $$ where $A$ is a symmetric, positive definit matrix. Let us denote the error by $e_0$; $$ e_0 := x - x_0 \,. $$ Furthermore, let the residual $$ r_0 := b - A x_0 \,. $$

We now want to make a correction step of the form $$ x_1 = x_0 + p \,, $$ to improve our approximation, where $p$ is some vector that we still need to determine. The error after the correction $e_1$ is $$ e_1 = x - x_1 = x - (x_0 + p) = e_0 - p \,. $$ We take a look at the $A$-norm of the error; we have that \begin{align*} \| e_0 - p \|_A^2 &= \dotp{A e_0 - A p}{e_0 - p} \\ &= \dotp{A e_0}{e_0} - \dotp{A e_0}{p} - \dotp{A p}{e_0} + \dotp{A p}{p} \\ &= \dotp{A e_0}{e_0} - 2 \dotp{Ae_0}{p} + \dotp{Ap}{p} \\ &= c - 2\dotp{r_0}{p} + \dotp{Ap}{p} \,, \end{align*} where we have used that $A e_0 = r_0$ and where $c$ is some constant independent of $p$. Let us define the function $\phi$ by setting $$ \phi(p) = \dotp{Ap}{p} - 2 \dotp{r_0}{p} + c \,. $$ The function $\phi$ is just the error of $e_1$ in dependence of the correction $p$ measured in the $A$-norm. Hence, we want to choose $p$ such that $\phi(p)$ is small. For this purpose, we compute the gradient of $\phi$.

A small computation which uses the symmetry of $A$ gives that $$ \nabla \phi(p) = 2(Ap - r_0) \,. $$ Recall that the negative of the gradient always points in the direction in which the function decreases the most. At position $x_0$, where $p=0$, we have that $$ \nabla \phi(0) = -2r_0 \,. $$ Thus, the function $\phi$ decreases the most in the direction of the residual $r_0$. This direction, is also called the direction of steepest decent.

For this reason, assume that we choose $p = \alpha_0 r_0$ for some scalar $\alpha$. In the next step, we choose, again, a vector $p$ and add its value to the current iterate. Using the same argument we choose $\alpha_1 r_1$ for some scalar $\alpha_2$. We continue in this manner. Thus, we have that $$ x_{k+1} = x_0 + \alpha_0 r_0 + \alpha_1 r_1 + \cdots + \alpha_k r_k \,, $$ in other words that $$ x_{k+1} \in x_0 + \mbox{span}\{ r_0, r_1, \dots, r_k \} \,. $$

Now, using the relation that $$ r_{k+1} = b - A x_{k+1} = b - A (x_k + \alpha_k r_k) = r_k - \alpha_k A r_k \,, $$ we see that $$ \mbox{span}\{ r_0, r_1, \dots, r_k \} = \mbox{span}\{ r_0, A r_0, \dots, A^{k-1} r_0 \} \,. $$ Hence, a Krylov subspace is the space spanned by the vectors of successive directions of steepest decent.

If you want to read more about a derivation of the conjugate gradient method which uses the above mentioned minimization principle, see An Introduction to the Conjugate Gradient Method Without the Agonizing Pain by J. R. Shewchuk.

Strictly speaking Krylov methods do not always work well. For instance it is possible to construct $A \in \mathbb{R}^{n \times n},b \in \mathbb{R}^n$, $A$ invertible, $b$ nonzero such that the first $n-1$ GMRES solutions to $Ax=b$ all have the same residual, and the residual only goes to zero at the $n$th step. This situation turns out to be rather pathological, though (i.e. you have to construct the problem from scratch, real world problems don't look like this).

Getting back to more realistic situations, you can use other bases as you see fit. But by directly studying Krylov methods, which are mathematically convenient, you can find out when they perform well (by proving convergence estimates), and then divide a general problem between "convert the problem to a problem where Krylov methods perform well" and "use a Krylov method". That first step is called preconditioning; there are lots of preconditioners for Krylov methods out there.