what is all solutions of X3 of the system ( b) [5 marks] Let X= 2...
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[47 b) [5 marks] Let X1 =12] and Let X, = 5 be two solutions of the linear [3] [6] system AX = B. Find all solutions X3 of this system, such that X3 + X, and X3 + X:. .-I .
The vectors 3 X3 =|-6|+t|4 X2 х, =|-2|+ t 12 are solutions of a system X' = AX Determine whether the vectors form a fundamental set of solutions.
(2 points) Let -2 3 -1 -6 9 -3 Describe all solutions of Ax = 0. x = x2 +X4 +x3
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by...
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5) Find all solutions to x3 + x2 + 2:-3 in F5 6) Find a solution to the equation 13-5 in Z/30Z
3. Two solutions of the following linear equation system are x1, X2, where Xi = (1,1,-3,1), x2-x1 + xd xd that makes cTx2 - cTx1 - 1, where c [1 1 2 1] Find every Ax=11 2 2 3 |x=b
3. Two solutions of the following linear equation system are x1, X2, where Xi = (1,1,-3,1), x2-x1 + xd xd that makes cTx2 - cTx1 - 1, where c [1 1 2 1] Find every Ax=11 2 2 3 |x=b
e) The temparature at the point y,z) is given by T(x,y,z) x2yz °C Use the method of tagrane multipliers to find the hottest and coldest points on the surface of the sphere x2y2z2 12. What are the hottest and coldest temperatures on the surface of the sphere in degrees Celsius? Question 2. (6 marks+ 4 marks+ 2 marks+3 marks+5 marks 20 marks) a) Find all solutions of the system of linear equation Ax = b where 2 3 12 5...
4. Let B = {x6 + 3, x5 + x3 + 1, x4 + x3, x3 + x2} C Pg, where Pg is the polynomials of degree < 8. (a) (2 marks) Explain why B is a linearly independent subset of Pg. (b) (2 marks) Extend B to a basis of Pg by adding only polynomials from the standard basis of Pg.
11. Let Z = (X1,X2, X3)T be a portfolio of three assets. E(X) 0.50. E(X2-1.5. E(X3) = 2.5, VAR(X)-2, VAR(X2)-3, Var(Xs)-5·PX1.x2-0.6 and X1 and X2 are idependent of X3 (a) Find E(0.3xi +0.3X2 +0.4X3) and Var(0.3X1 +0.3X2 +0.4Xs) (b) Find P[0.3X1 +0.3x2 + 0.4X3 <2). Since z-table isn't provided, just write down the (c) Find the covariance between a portfolio that allocates 1/3 to each of the three assets and a portfolio that allocates 1/2 to each of the first...