Transcription of 17. Image Measure - Probability
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17. Image Measure - Probability
Tutorial 17: Image Measure 1. 17. Image Measure In the following, K denotes R or C. We denote Mn (K), n 1, the set of all n n-matrices with K-valued entries. We recall that for all M = (mij ) Mn (K), M is identi ed with the linear map M : Kn Kn uniquely determined by: n .. j = 1, .. , n , M ej = mij ei i=1. i . where (e1 , .. , en ) is the canonical basis of Kn , ei = (0, ., 1 , ., 0). Exercise 1. For all K, let H Mn (K) be de ned by: .. 1 0 .. H = .. 0 .. 1. Tutorial 17: Image Measure 2. by H e1 = e1 , H ej = ej , for all j 2. Note that H is obtained from the identity matrix, by multiplying the top left entry by . For k, l {1, .. , n}, we de ne the matrix kl Mn (K) by kl ek = el , kl el = ek and kl ej = ej , for all j {1.}
Tutorial 17: Image Measure 5 6. Show that multiplying M by Σkl from the left, amounts to in- terchanging the rows Rl and Rk. 7. Show that multiplying M by Σkl from the right, amounts to interchanging the columns Cl and Ck. 8. Showthatmultiplying M by U−1 from the left (n ≥ 2),amounts to subtracting R1 from R2, i.e.: U−1. R1 R2 Rn R1 R2 −R1 Rn 9. Show that multiplying M by U−1 from ...
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