Linear Algebra: Graduate Level Problems and Solutions

Linear Algebra Igor Yanovsky, 2005 4 1 Basic Theory 1.1 Linear Maps Lemma. If A 2 Matmxn(F) and B 2 Matnxm(F), then tr(AB) = tr(BA): Proof. Note that the (i;i) entry in AB isPn P j=1 fiijflji, while (j;j) entry in BA is m i=1 fljifiij. Thus tr(AB) = Xm i=1 Xn j=1 fiijflji; tr(BA) = Xn j=1 Xm i=1 fljifiij: 1.2 Linear Maps as Matrices

  Linear, Theory

The Lebesgue integral - MIT Mathematics

36 2. THE LEBESGUE INTEGRAL Proof. These are basic properties of the Riemann integral see Rudin [2]. Note that C c(R) is a normed space with respect to kuk L1 as de ned above. With this preamble we can directly de ne the ‘space’ of Lebesgue integrable