How to spot this family

• M2,2 subspace + matrix of a linear map："S = {A ∈ M2,2 : ... and tr(A)=0}" · "T: M2,2 → P1 ... every matrix of T must be 2×4" · 矩阵空间上的双约束子空间 + 迹型线性映射
• P2 subspace + matrix of a linear map："Find a basis for S = {p ∈ P2 : ...}" · "find the matrix of T with respect to the standard bases" · 多项式空间 $\mathbb{P}_2$ 上的约束子空间 + 求值型线性映射
• Change of basis & coordinates："coordinates of x with respect to the basis B" · "change of basis matrix" · 给非标准基求坐标/转换矩阵
• Subspace proof, basis & dimension："show that V is a subspace ... find a basis and the dimension" · 约束形如 $x_1-x_3+5x_4=0$ 或 $p(1)=2p(-1)$
• Linear map: kernel basis & rank："show that T is linear ... find a basis for ker(T) and the rank" · $T(\mathbf{x})$ 是分量的线性组合