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Unformatted text preview: Practice Test 2 For full credit, show all work. 1 State the definition for the following: a) A basis for a subspace S. b) An invertible matrix M . c) rank(A). 3 1 4 6 2 Given A = 2 0 1 3. Find the following: -1 1 2 0 a) A basis for null(A). b) A basis for column(A). 3a) Show that projection onto y = 2x is a linear transformation. b) Find the standard matrix representation for the projection. 4 Find counter-examples to the following false statements: each) a) (A + B)-1 = A-1 + B -1 . b) AB = AC = B = C. (10 pts 2 1 -2 5a) Find the LU factorization for A = -2 3 -4. 4 -3 0 -3 b) Use the LU factorization to solve the sytem Ax = 1 . 0 6 Suppose the weather of a city is a Markov process. The probability that tomorrow is dry is 8/10 if today is dry, and 4/10 if today is wet. The probability that tomorrow is wet is 2/10 if today is dry, and 6/10 if today is wet. a) Write down the transition matrix P for the above Markov process. Show that the matrix is stochastic. b) What is the distribution of wet and dry days in the long run? 7 Prove 2 of the following statements (do the remaining one for extra credit): a) If A and B are square matrices and AB is invertible, then both A and B are invertible. b) If A and B are n n matrices of rank n, then AB has rank n. c) If R is a matrix in echelon form, then a basis for row(A) consists of the non-zero rows of R. ...
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