miu=121, lemda =1 Q2[20]. Let A = (3,2, A), B (5, d, 2). Compute (a) A+B (b) A B (c) ||A|| (d) 3A (e) ||A- B|| Q2[20]. Let A = (3,2, A), B (5, d, 2). Compute (a) A+B (b) A B (c) ||A|| (d) 3A (e) ||A- B||
Let W = span{ (-2, 1; 2,0), (4,0-5, 2)] the Groun - Student Produrt to find an enthonormal basis for W The dimension of wt, the orthogonal complement of w.is basis from (a), either before or after normalizing projw (1,1,1,). be your to find
1. Consider the following three vectors in R Vi (1,-1,-11), v2 (3,0,-3,2), v3- (4,0,-2,2) (a) Perform the Gram-Schmidt process to find an orthonormal basis [ei,e2,e3j of the subspace spanned by {vi, V2, V3) (b) Find the QR decomposition of the following matrix A QR: 412 922 231 12 113 q13 q23 43300 14 924 934 -1 0 0 0 122 r23 Relate (rij] to the Gram-Schmidt process. (c) Can you say anything about either Qor without calculation? Show that ATA...
Let B={(4,0), (0,3)} and v = (12,6). Find [v]_B, the coordinate vector of v, relative to basis B. (To enter a height 2 column vector, use the notation (a,b)^T.)
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you wish to best-fit these to a line y = mx + b using least-squares. (a) Write down the corresponding matrix equation. (b) Solve for using the method of least squares. Make sure you simplify: the answer should not be complicated. (c) Find limin (d) The line corresponding to your answer in (c) passes through (3,2). Why does this make sense?
TI (5,5) | (3,10) (0,4) M (10,3) (4,4)2,2) B (4,0) (2,-2) (-10,-10) Let G(8) denote the game in which the game G is played by the same players at times 0, 1,2,3,... and payoff streams are evaluated using the common discount factor a. For which values of is it possible to sustain the vector (5, 5) as a subgame per- fect equilibrium payoff, by using Nash reversion (playing Nash eq. strategy infinitely upon a deviation) as the punishment strategy.
Let X~N(1,2) and Y~N(3,2). Define the distribution of 2(X-Y).
Question 1 Let A 6 b- > [|1 and C 5 -2 3. Calculate the following: 4 0 (a) 2(A - 30) (6) ACT CC A(4b) Only work on your blank sheets of paper. You will submit your work a
please 2 only, thanks Exercises dA (1) Use Cauchy's residue theorem to compute Jo 2+sin (2) Repeat the preceding exercise for 8" 131. (3) Let a be a complex number such that lal < 1. Prove that (2 27 Jo 1 - 2a cos 0 + a2d6 = 1 - 22 (4) What is the value of the integral in the preceding exercise when |al > 1? (Hint: Let b= 1.)
4. Let T: R - R be a linear transformation such that 2x1 32 32 3 -1 22 + T3 T 2 Find the standard matrix of T. 5. Compute 3A - 4B, AB and BA: 021 3 1 0 and B -2 0 1 3 2 1 A = 1 -4 1 1 4 0 6. Find the inverse of each matrix BEE 0 1 -2 4 -6 and 1 1 3 -5 3 -1 1