Basic Concepts

Affine Set

Definition

\(C \subset \mathbb{R}^n\) is affine iff \(\forall x_1, x_2 \in C, \theta x_1 + \left(1 - \theta\right) x_2 \in C\)

Subspace

Solution Set of Linear System

Affine Hull

Zero Norm

Zero norm is the number of non-zero elements in a matrix, which is used to evaluate the sparsity of the matrix.

Convex Set

Simplex

For \(k+1\) affinely independent ¹ points \(\boldsymbol{v}_0,\boldsymbol{v}_1,\boldsymbol{v}_2,\cdots,\boldsymbol{v}_k \in \mathbb{R}^n\;(k \le n)\), the simplex defined by these points are

\[\mathrm{conv}\;\{\boldsymbol{v}_0,\boldsymbol{v}_1,\boldsymbol{v}_2,\cdots,\boldsymbol{v}_k\} = \left\{\left.\sum_{i=0}^k\theta_i\boldsymbol{v}_i \right| \theta_i \gt 0, \sum_{i=0}^k\theta_i=1\right\}\]

单纯形是仿射无关的点集的凸包，任意单纯形都是多面体。

脚注

1. \(\boldsymbol{v}_0,\boldsymbol{v}_1,\boldsymbol{v}_2,\cdots,\boldsymbol{v}_k\) are affinely independent iff \(\boldsymbol{v}_1 - \boldsymbol{v}_0, \boldsymbol{v}_2-\boldsymbol{v}_0, \cdots, \boldsymbol{v}_k - \boldsymbol{v}_0\) are linearly independent.