T invariant subspace #linear algebra

T-Invariant Subspace: A subspace W \subseteq V where T(w) \in W for every w \in W. Basis Extension Theorem: Constructing a basis \beta = \{v_1, \dots, v_k, v_{k+1}, \dots, v_n\} for V by extending an ordered basis \{v_1, \dots, v_k\} of W. Block Upper Triangular Matrix: Because T(v_j) \in W for all j \le k, the first k columns of [T]_\beta have zeros in the bottom n-k entries, forming the zero matrix O_{(n-k) \times k}. Hashtags: