fa11_mae260_hw4 - MAE 260. Homework Assignment #4 Issued:...

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Unformatted text preview: MAE 260. Homework Assignment #4 Issued: Nov. 10, 2011 Due: Nov. 24, 2011 (in class) 1. The trace of a square matrix A Rnn , denoted as Tr(A) stands for the sum of all the diagonal elements: Tr(A) = Show that: (a) Tr(AB) = Tr(BA). (b) Tr(ABC) = Tr(BCA) = Tr(CAB). n Aii . i=1 2. Come up with matrix A such that (a) A 0 but A2 = 0. (b) A2 0 but A3 = 0. 3. Suppose you solve Ax = b for three special right sides b: 0 0 1 Ax1 = 0 , and Ax2 = 1 , and Ax3 = 0 1 0 0 4. Find all matrices A = a b 1 1 1 1 that satisfy A = A. c d 1 1 1 1 5. Show that matrices A and [A AB] (with extra columns) have the same column space. (Hint: you need to show that every column of AB can be expressed as a linear combination of columns of A.) 6. Show that the rank of X Rnn defined by X = xxT for some x Rn is one. 7. For linear equation Ax = b with x Rn and b Rn+1 . Show that Ax = b has no solution if matrix [A b] is invertible. 8. Prove that if a = 0 or d = 0 or f = 0, the columns of U are dependent: a b c U= 0 d e 0 0 f 1 ...
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