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。