Machine learning--数学相关（2）
Vector Spaces(向量空间)
Groups（群）
Consider a set \(\mathcal{G}\) and
an operation \(\otimes: \mathcal{G} \times
\mathcal{G} \rightarrow \mathcal{G}\) defined on \(\mathcal{G}\). Then \(G:=(\mathcal{G}, \otimes)\) is called a
group if the following hold:
1. Closure of \(\mathcal{G}\) under
\(\otimes: \forall x, y \in \mathcal{G}: x
\otimes y \in \mathcal{G}\)
2. Associativity: \(\forall x, y, z \in
\mathcal{G}:(x \otimes y) \otimes z=x \otimes(y \otimes
z)\)
3. Neutral element: \(\exists e \in
\mathcal{G} \forall x \in \mathcal{G}: x \otimes e=x\) and \(e \otimes x=x\)
4. Inverse element: \(\forall x \in
\mathcal{G} \exists y \in \mathcal{G}: x \otimes y=e\) and \(y \otimes x=e\), where \(e\) is the neutral element. We often write
\(x^{-1}\) to denote the inverse
element of \(x\).
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