Question

Assignment description Let A = _ au (221 012] 1. The trace of A, denoted tr A, is the sum of the diagonal entries of A. In other words, 222 ] tr(A) = 211 + A22 For example, writing 12 for the 2 x 2 identity matrix, tr(12) = 2. Submit your assignment © Help Q1 (1 point) Let V be a vector space and let T : M2x2(R) → V be a non-zero linear transformation such that T(AB) = T(BA) for all A, BE M2x2(R). Show that (a) T(A) = ({tr A)T(12) for all A € M2x2(R). (b) dim(ker T) = 3. Hint: A = (A - (Ztr A)12) + (4tr A)12.

Answer 1

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wher hz Assume k#3, (Ex;) = T( Eus Eij) 27 ( Eik. Ejx) ICAL) 2 T (B1) - f@ J=0 - }(Exi) - TI EKK) = T( Eki Elu) ST(Eik Eni] of follows that 2T ( Eu) T(Eis) 20 * i tj - zT (F.) Nizi NOW IT Ê tii: T(I) = ¿ Tebii) = n. T(eu) T(EU) - FC T( fii) 2 TCP) now consider trace tinction for ren mabies to (Eis) zo i #j, To (Eij) z 1 if izj T(EI) - FC to (Bis). fon 122,6 T(A)= FCH) tv (A) = { trea). TCP) kertaah ! TCA)=0 z'ff. dr(a) zom idem kert = 4-123. n to(A) Zo