# Diagonalizability and trigonalizability of matrices over complex numbers ### Core idea - A set with an internal addition law (+) and an external multiplication law (x) by real numbers is a real vector space if it satisfies specific axioms for group properties, - A subset F of a vector space E is a subspace if it is closed under E's addition and external multiplication, meaning for any x, y in F and any ### Key facts * Properties derived from vector space definition include $0 \times x = 0$, $(-1) \times x = -x$, $t \times 0 = 0$, and $tx = 0 \Leftrightarrow t = 0$ or $x = 0$ [5](#page=5).