Mathematics, one idea at a time.

A vector space over a field is a set of vectors that can be added and multiplied by scalars, with the usual rules of arithmetic. Vector spaces are the main objects of linear algebra.

The ring of integers $\mathbb{Z}$ is a Euclidean domain, with the absolute value $\delta(n) = |n|$ as Euclidean function. That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$, there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$.

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ if and only if the following holds. For each sequence $(x_n)$ in $D \setminus \{a\}$ with $x_n \to a$, $f(x_n) \to L$.

Let $A$ be an $m \times n$ complex matrix. Then there are a unitary matrix $U$ of size $m \times m$, a unitary matrix $V$ of size $n \times n$, and an $m \times n$ matrix $\Sigma$ with $$A = U \Sigma V^*.$$ Here $\Sigma_{ij} = 0$ for $i \ne j$, and $\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0$ are the singular values of $A$.

The quotient set of a set by an equivalence relation is the set of all equivalence classes.

A family of events is mutually independent when, for each finite subfamily, the probability that all its events occur is the product of their probabilities. The node mutual independence of events gives the precise definition.