Let T: R2 -> R2 be a linear
transformation defined by T(x1 , x2) =
(x1 + 2x2 , 2x1 +
4x2)
a. Find the standard matrix of T.
b. Find the ker(T) and nullity (T).
c. Is T one-to-one? Explain.
Consider the linear system of equations below
3x1 − x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4
i. Use the Gauss-Jacobi iterative technique with x
(0) = 0 to find
approximate solution to the system above up to the third step
ii. Use the Gauss-Seidel iterative technique with x
(0) = 0 to find
approximate solution to the third step
3. Given is the function f : Df → R with F(x1, x2, x3) = x 2 1 +
2x 2 2 + x 3 3 + x1 x3 − x2 + x2 √ x3 . (a) Determine the gradient
of function F at the point x 0 = (x 0 1 , x0 2 , x0 3 ) = (8, 2,
4). (b) Determine the directional derivative of function F at the
point x 0 in the direction given...
Consider the following LP model.Max Z = 3x1 - 4x2 +
x3
subject to x1 + x2 + x3 >= 9
2x1
+ x2 + x3<= 12
x1 + x2 = 5
x1, x2, x3 >= 0
Change it to standard form.
Obtain all the basic solutions and indicate which ones are basic
feasible solutions and write down the corresponding corner points.
For each basic solution, you have to obtain the values of all the
variables.
Obtain the solution of the LP...
Consider the following quadratic forms
q(x1, x2) = 3x1^2 − 6x1x2 + 11x2^2 and
r(x1, x2, x3) = x1^2 − x2^2+x3^2+ 2x1x2 − 6x1x3+2x2x3,
on R 2 and R 3 , respectively. In both cases do the
following.
(a) Find the symmetric matrix A representing the quadratic
form.
(b) Find a corresponding orthogonal matrix P of eigenvectors of
that matrix.
(c) Write down the maximum and minimum values of the quadratic
form over the unit vectors (in R 2 and...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then, find a two dimensional sufficient statistic for (a, b)
Consider the linear system of equations
2x1 − 6x2 − x3 = −38
−3x1 − x2 + 7x3 = −34
−8x1 + x2 − 2x3 = −20
With an initial guess x (0) = [0, 0, 0]T solve the system using
Gauss-Seidel method.
let X1, X2, X3 be random variables that are defined as
X1 = θ + ε1
X2 = 2θ + ε2
X3 = 3θ + ε3
ε1, ε2, ε3 are independent and the mean and variance are the
following random variable
E(ε1) = E(ε2) = E(ε3) = 0
Var(ε1) = 4
Var(ε2) = 6
Var(ε3) = 8
What is the Best Linear Unbiased Estimator(BLUE) when estimating
parameter θ from the three samples X1, X2, X3