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 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