Math 375 - 11/8

Effect of Row Operations on Determinants Row Operation Determinant Row A→Row A+c⋅Row B det⁡M→det⁡M Row A→c⋅Row A det⁡M→c⋅det⁡M Row A↔︎┴switch Row B det⁡M→−det⁡M Understanding of Matrix Multiplication in terms of Linear Map Composition • Motivation ○ V→┴T W→┴S Z • Setup ○ {e_1…e_n }: basis of V ○ {f_1…f_m }: basis of W ○ {g_1…g_k }: basis of Z ○ Let m(T)=(t_ij ) ○ Let m(S)=(s_ij ) • Claim ○ m(S)⋅m(T)=m(ST) • Proof ○ T(e_i )=∑_(j=1)^m▒〖t_ij f_j 〗 ○ S(f_j )=∑_(k=1)^r▒〖s_jk g_k 〗 ○ ST(e_i )=∑_(j=1)^{m▒∑_(k=1)}r▒〖t_ij s_jk g_k 〗 ○ Which is the same as matrix multiplication