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Disc10 - Math 17B Kouba Discussion Sheet 10 0 1 —1 0 RX...

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Unformatted text preview: Math 17B Kouba Discussion Sheet 10 0 1 —1 0 RX and plot X and RX on the same mag-coordinate sytem. a.)X=(§) b.>X=(3) C-)X=(—03> d')X=(-11) e.) Make a conjecture about how vector RX is related to vector X . . . _ fi/2 —1/2 . . . . 7 2.) ConSIder the matrix R — < 1/2 x/g/2 . Is this matrix a rotation matrix . If so, in what direction and how many radians does it rotate vectors ? 1 0 0 1 Solve the matrix equation det(A — /\I) z 0 for A for each of the following matrices A. a.)A=(: 31> b.)A=<i: 3:50) c.)A=<_35 i) d-)A=(i) 12) 4.) Find eigenvalues and the corresponding eigenvectors for each matrix. a» (a a) b» (1 3) c» <3: 24) 1 2 1 0 corresponding to distinct eigenvalues. 5 7 1.) Consider the matrix R :2 ( > . For each of the following vectors X, compute 3.) Let I = < > be the 2:102 identity matrix and let /\ be a constant (real or imaginary). 5.) Consider the matrix A = ( > from problem 4.)b.) with eigenvectors V1 and V2 a.) Write the vector ( > as a linear combination of V1 and V2 , i.e., determine constants c1 and C2 so that 01V1 + Cng = (g). b.) Use your results in part a.) to compute A20 <3) . 6.) Following are Leslie matrices. Find both eigenvalues, determine if the population is increasing or decreasing, and find the stable age distribution for each matrix. ‘1‘), (oh 3) b‘) (0231 (1)) “ Wisdom outweighs any wealth.”— Sophocles ...
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