The below is the digested summary of the Krylov properties section from this book. My question is about the conclusion presented at the end.
By applying the Cayley-Hamilton theorem, for $A \in \mathbb{R}^{n \times n}$ and $\forall y \in \mathbb{R}^n$, we can derive that
$$ A^{-1}y = -\frac{1}{|A|}\sum_{i=0}^{n-1}c_{i+1}A^iy $$
Now, assuming $|A| \ne 0$, let $x = A^{-1}b$ be the solution to the linear system $Ax = b$.
Let $x^{(0)}$ be any arbitrary vector. We define $e^{(0)} = x - x^{(0)}$, and $r^{(0)} = b - Ax^{(0)} = Ae^{(0)}$. Substituting $y = r^{(0)}$ in the above equation, we have
$$ e^{(0)} = A^{-1}r^{(0)} = -\frac{1}{|A|}\sum_{i=0}^{n-1}c_{i+1}A^ir^{(0)} $$
I understand everything thus far. However, the next part confuses me, wherein the authors state:
We conclude that: $$ A^{-1}b \in x^{(0)} + \mathcal{K}_n(A,r^{(0)}) $$ where $$ \mathcal{K}_n(A,r^{(0)}) = \text{span}(r^{(0)}, Ar^{(0)}, A^2r^{(0)}, \dots, A^{n-1}r^{(0)}) $$ is the $n^\text{th}$ Krylov subspace associated to $A$ and $r^{(0)}$. $$
I just do not understand the last part of the derivation i.e. how did they come to this conclusion? Shall appreciate a bit more details explaining this.
From what you've said you understand, you've got \begin{align} A^{-1}b&=x\\ &=x^{(0)}+e^{(0)}\\ &=x^{(0)}-\frac{1}{|A|}\sum_{i=0}^{n-1}c_{i+1}A^ir^{(0)}\\ &\in x^{(0)}+\text{span}(r^{(0)}, Ar^{(0)}, A^2r^{(0)}, \dots, A^{n-1}r^{(0)})\ . \end{align}