Math 307 Spring, 2005 Hentzel Wednesday, March 30, 2005 Test III, 1. For the Markov Chain, find the steady state. ______ .1 ______ | | ------------> | | | A | <---.3------ | B | |_____| __ __ |____| \ /\ /| / .1 (.2) (.1) .1 \ \ / / \ \ / / _\| \ / |/_ __________ | C | |________| If at the start (A,B,C) = {100,100,100} what happens after 2 instances of time? A B C A .8 .3 .2 B .1 .6 .1 C .1 .1 .7 | -.2 .3 .2 | | .1 -.4 .1 | | .1 .1 -.3 | | -2 3 2 | | 1 1 -3 | | 1 0 -11/5 | 1/10 | 1 -4 1 | == | 0 -5 4 | == | 0 1 -4/5 | | 1 1 -3 | | 0 -5 4 | | 0 0 0 | | 11 | The steady state is | 4 | . | 5 | Check | .8 .3 .2 | | 11 | | 8.8+1.2+1.0 | | 11 | | .1 .6 .1 | | 4 | = | 1.1+2.4+ .5 | = | 4 | It checks. | .1 .1 .7 | | 5 | | 1.1+ .4+3.5 | | 5 | Part (b) 0 1 2 | .8 .3 .2 | | 100 | | 130 | | 104+24+18 | | 146 | | .1 .6 .1 | | 100 | ; | 80 | ; | 13+48+ 9 |== | 70 | | .1 .1 .7 | | 100 | | 90 | | 13+ 8+63 | | 84 | ------------------------------------------------------------------- 2. Find the change of basis matrix P for the bases BC | 1 2 0 | |8 1 1 | B = | 0 0 1 | C = |1 8 0 | | 0 1 0 | |1 1 8 | V = B Vb = C Vc -1 Vb = B C Vc -1 -1 | 1 2 0 | | 8 1 1 | Pbc = B C = | 0 0 1 | | 1 8 0 | | 0 1 0 | | 1 1 8 | | 1 0 -2 | | 8 1 1 | = | 0 0 1 | | 1 8 0 | | 0 1 0 | | 1 1 8 | | 6 -1 -15 | = | 1 1 8 | | 1 8 0 | Pbc | x | Vc = | y | | z | | 6x-y-15z | Vb = | x+y +8z | | x+8y | |8x+y+z | | 8x+y+ z | V = C Vc = | x+8y | = B Vb = | x+8y | |x+y+8z | | x+y+8z | ------------------------------------------------------------ | 1 2 0 | 3. Suppose T = | (8) (8) 1 | BB | 1 0 (8) | | 1 0 1 | where the basis B is | 0 1 1 | | 1 0 0 |. | 1 1 0 | Find the matrix T for the basis C is | 0 0 1 |. CC | 1 0 0 | -1 -1 C B Tbb B C Vc = Wc -1 -1 | 1 1 0 | | 1 0 1 | | 1 2 0 | | 1 0 1 | | 1 1 0 | | 0 0 1 | | 0 1 1 | | 8 8 1 | | 0 1 1 | | 0 0 1 | | 1 0 0 | | 1 0 0 | | 1 0 8 | | 1 0 0 | | 1 0 0 | | 0 0 1| | 1 0 1 | | 1 2 0 | | 0 0 1| | 1 1 0 | | 1 0 -1| | 0 1 1 | | 8 8 1 | |-1 1 1| | 0 0 1 | | 0 1 0| | 1 0 0 | | 1 0 8 | | 1 0 -1| | 1 0 0 | | 1 0 0 | | 1 2 0 | | 1 0 0| | 0 0 1 | | 8 8 1 | | 0 -1 1| | 0 1 1 | | 1 0 8 | | 0 1 0| | 1 2 0 | | 1 0 0| | 1 0 8 | | 0 -1 1| | 9 8 9 | | 0 1 0| | 1 -2 2 | | 1 8 0 | | 9 1 8 | CC = {{1,1,0},{0,0,1},{1,0,0}}; B = {{1,0,1},{0,1,1},{1,0,0}}; Tbb = {{1,2,0},{8,8,1},{1,0,8}}; Inverse[CC].B.Tbb.Inverse[B].CC Out[5]//MatrixForm= 1 -2 2 1 8 0 9 1 8 4. Solve the following differential equation. / | x | | 3 -4 0 4 | | x | | y | = | -2 3 0 2 | | y | | z | | 0 0 9 0 | | z | | w | | -2 -4 0 9 | | w | | 1 0 1 0 | | 1 1 0 0 | P = | 0 0 0 1 | | 1 1 1 0 | | 1 1 0 -1 | -1 | -1 0 0 1 | P = | 0 -1 0 1 | | 0 0 1 0 | -1 P A P = | 1 1 0 -1 | | 3 -4 0 4 | | 1 0 1 0 | | -1 0 0 1 | |-2 3 0 2 | | 1 1 0 0 | | 0 -1 0 1 | | 0 0 9 0 | | 0 0 0 1 | | 0 0 1 0 | |-2 -4 0 9 | | 1 1 1 0 | -1 P A P | 1 1 0 -1 | | 3 0 7 0 | | -1 0 0 1 | | 3 5 0 0 | | 0 -1 0 1 | | 0 0 0 9 | | 0 0 1 0 | | 3 5 7 0 | | 3 0 0 0 | | 0 5 0 0 | | 0 0 7 0 | | 0 0 0 9 | | x | | 1 0 1 0 | | c1 e^(3t) | | y | | 1 1 0 0 | | c2 e^(5t) | | z | = | 0 0 0 1 | | c3 e^(7t) | | w | | 1 1 1 0 | | c4 e^(9t) | Check / | x | | 1 0 1 0 | | 3 c1 e^(3t) | | 3 0 7 0 | | c1 e^(3t) | | y | | 1 1 0 0 | | 5 c2 e^(5t) | | 3 5 0 0 | | c2 e^(5t) | | z | = | 0 0 0 1 | | 7 c3 e^(7t) | = | 0 0 0 9 | | c3 e^(7t) | | w | | 1 1 1 0 | | 9 c4 e^(9t) | | 3 5 7 0 | | c4 e^(9t) | | x | | 3 -4 0 4 | | 1 0 1 0 | | c1 e^(3t) | | 3 0 7 0 | | c1 e^(3t) | A.| y | = |-2 3 0 2 | | 1 1 0 0 | | c2 e^(5t) | = | 3 5 0 0 | | c2 e^(5t) | | z | | 0 0 9 0 | | 0 0 0 1 | | c3 e^(7t) | | 0 0 0 9 | | c3 e^(7t) | | w | |-2 -4 0 9 | | 1 1 1 0 | | c4 e^(9t) | | 3 5 7 0 | | c4 e^(9t) | A 5. Tidbits: (a) Fill in the bottom row to make this matrix stochastic. | .5 .2 (.8)| | .3 (.8) .1 | | .2 0 .1 | (b) If B and C are basis and T is a linear transformation, BB Express T in terms of B, C and T . CC BB Tcc = C^(-1) B Tbb B^(-1) C (c) What is the dimension of / | 1 | | 2 | | 3 | | 4 | | 5 | \ / | 6 | | 7 | | 8 | | 9 | |10 | \ / |11 | |12 | |13 | |14 | |15 | \ \ |16 |, |17 |, |18 |, |19 |, |20 | / \ |21 | |22 | |23 | |24 | |25 | / \ |26 | |27 | |28 | |29 | |30 | / Spanned by the first vector and the all ones vector. Rank = 2 (d What is the process to find the steady state of a stochastic matrix B? Solve (A-I)X = 0. (e) Prove that the product of two stochastic matrices also stochastic? Let X be the all one's row vector and A and B two stochastic matrices. Then X(AB) = (XA)B = XB = X. 4 (f) Give a spanning set of R which is not a basis. | 1 | | 0 | | 0 | | 0 | | 1 | | 0 | | 1 | | 0 | | 0 | | 1 | | 0 |, | 0 |, | 1 |, | 0 |, | 1 | | 0 | | 0 | | 0 | | 1 | | 1 | (g) What is the Null Space of a matrix. The set of all X such that AX = 0. (h) What is the rank of a matrix? The number of non zero rows in the row canonical form.