In recent years, many fields have involved the problem of solving large-scale sparse systems of equations. When the problem size is extremely large, solving them becomes very difficult. Fortunately, Krylov subspace methods can handle such problems very well, and thus they have also been rated as one of the ten greatest algorithms of the 20th century. In this article, we will introduce the basic idea of the Krylov subspace and the famous GMRES algorithm (Generalized Minimal Residual Method).

The Weierstrass Approximation Theorem tells us that any continuous function $f$ on a closed interval can be uniformly approximated by polynomial functions. In 1912, Bernstein, based on probability theory, constructed Bernstein polynomials and proved the Weierstrass Approximation Theorem. This paper will introduce the essence of how Bernstein polynomials can approximate continuous functions from both the probabilistic and pure analytic perspectives. First, we present the Weierstrass Approximation Theorem:

Weierstrass Approximation Theorem

Theorem 1 (Weierstrass Approximation Theorem): Let $f$ be a continuous function on the closed interval $[a,b]$. For any $\varepsilon > 0$, there exists a polynomial $p$ such that

$$
\max_{x \in [a,b]} |p(x) - f(x)| < \varepsilon
$$

Proposition 1. Let $E=F(u)$, where $u$ is transcendental over $F$. If $K\neq F$ and $K$ is an intermediate field of $E/F$, then $u$ is algebraic over $K$.

Theorem 1. Let $E/F$ be a field extension, and let $a$ be algebraic over $F$. Then
$$
[F(a):F]=\deg f(x),
$$
where $f(x)$ is the minimal polynomial of $a$ over $F$.

Proposition 1. The only automorphism of the rational field $\mathbb{Q}$ is the identity automorphism.

Definition. Let $R$ be a commutative ring with identity. If $I$ and $J$ are two ideals of $R$ such that
$$
I+J=R,
$$
then $I$ and $J$ are said to be coprime.

Proposition 1. Let $I$ and $J$ be ideals of a ring $R$. Then:

1. $IJ \subseteq I$ and $IJ \subseteq J$.

2. $IJ \subseteq I \cap J \subseteq I+J$.

First, let us review the notion of an ideal generated by an element.

Theorem. Let $R$ be a ring with identity, and let $a,b \in R$. Then
$$
1-ab \text{ is invertible } \iff 1-ba \text{ is invertible}.
$$

Lemma 1. Let $R$ be a ring with identity, and let $a \in R$. Suppose that $a$ has a left inverse in $R$. Then the following are equivalent:

1. $a$ has more than one left inverse.

2. $a$ is not a unit.

3. $a$ is a right zero divisor of some element.