Exercise:
abcliste abc forall Ain M_mtimes nK Bin M_ntimes pK Cin M_ptimes qK we have: ABC ABC in M_mtimes qK. abc forall Ain M_mtimes nK Cin M_ntimes pK C'in M_qtimes mK we have: A+BC AC+BC in M_mtimes pK C'A+B C'A+C'B in M_qtimes nK. abc forall Ain M_mtimes nK we have I_mAA AI_nA abc forall alpha Ain M_mtimes nK Bin M_ntimes pK we have: alpha ABalpha ABAalpha B abcliste

Solution:
abcliste abc Let T_A:K^nlongrightarrow K^m T_B:K^plongrightarrow K^n T_C:K^qlongrightarrow K^p be the linear maps defined by ABC. i.e. T_AvAvquad forall vin K^n T_BwBwquad forall win K^p T_CuCuquad forall uin K^q forall r denote by epsilon_r the standard basis of K^r. We have seen that T_A_epsilon_m^epsilon_nAT_B_epsilon_n^epsilon_pB T_C_epsilon_p^epsilon_qC. Now T_Acirc T_Bcirc T_C T_Acirc T_Bcirc T_C K^prightarrow K^mK^qrightarrow K^mK^nrightarrow K^mK^qrightarrow K^n We now pass to the matrix representation of the above leftT_Acirc T_Bcirc T_Cright_epsilon_m^epsilon_q leftT_Acirc T_Bcirc T_Cright_epsilon_p^epsilon_q &Longrightarrow T_Acirc T_B_epsilon_m^epsilon_p T_C_epsilon_p^epsilon_q T_A_epsilon_m^epsilon_n T_Bcirc T_C_epsilon_n^epsilon_q &Longrightarrow leftT_A_epsilon_m^epsilon_n T_B_epsilon_n^epsilon_pright T_C_epsilon_p^epsilon_qT_A_epsilon_m^epsilon_n leftT_B_epsilon_n^epsilon_p T_C_epsilon_p^epsilon_qright &Longrightarrow A B C A B C. abc T_A+T_Bcirc T_C T_Acirc T_C +T_B circ T_C &Longrightarrow leftT_Acirc T_Bcirc T_Cright_epsilon_m^epsilon_p T_Acirc T_C + T_Bcirc T_C_epsilon_m^epsilon_p &Longrightarrow T_A+ T_B_epsilon_m^epsilon_n T_C_epsilon_n^epsilon_p T_Acirc T_C_epsilon_m^epsilon_p+T_Bcirc T_C_epsilon_m^epsilon_p &Longrightarrow A+B C A C + B C. The other identity in b is similar. abc I_mid_K^m_epsilon_m^epsilon_mquad id_K^mcirc T_A T_A &Longrightarrow id_K^mcirc T_A_epsilon_m^epsilon_nT_A_epsilon_m^epsilon_n A &Longrightarrow I_m AA. The other identity is similar. abc bf claim. Let V be a finite dimensional VS over K alpha in K. Let B be a basis for V. Consider the map Q:Vlongrightarrow V Qv: alpha v. Then V is linear and moreover we have Q_mathcalB^mathcalBalpha I_n where ntextdimV. to be proven. abcliste