1.1. (a) Manufacture a 4 £ 4 matrix A with eigenvalues at 5, 7, 13 and
17 and eigenvectors precisely at
v1 =(1011); v2=(1311); v3 = (235-2); v4 = (34-5-1)                              (1)
 (b) Orthogonalize the vectors v1, v2, v3, v4 using Gramm-Schmidt
Orthogonalization. Call the
orthogonal vectors u1, u2, u3, u4.
(c) Manufacture a 4 x 4 symmetric, positive definite matrix B with
eigenvalues at 5, 7, 13 and17 and eigenvectors precisely at u1, u2,
u3, u4.
(d) Consider the ellipsoid in IR4 given by ( x1 x2 x3 x4) B (x1x2x3x)=
100           (2)
Write down the homogeneous, quadratic, polynomial equation of the
ellipsoid using variables
x1, x2, x3, x4. Define a new set of variables y1, y2, y3, y4 such that
the ellipsoid is given as
5y1^2 + 7y2^2 + 13y3^2 + 174y2^ = 100
(e) Compute the point of intersection p between the line
L = {x1 = 3t; x2 = 4t; x3 = ¡5t; x4 = ¡t : t 2 IRg} and the ellipsoid (2).
(f) Compute equation of the tangent plane to the ellipsoid at the point p. 
(g) Consider a function
                                   Á(x1; x2; x3; x4) = 8x1 + 9x2 + 7x3 - 12x4:
Find maximum and minimum value of Á subject to the constraint that
(x1; x2; x3; x4) belongs
to the ellipsoid.
2.let us deifine X=(x1 x2 x3 x4 )
(a) Construct a 4x4 matrix A and a vector of initial conditions X0
such that when you solve
Xdot= AX; X(0) = X0
 we get x1 = sin2t, x3 = sin4t. Also calculate x2 and x4 for your
choice of A and X0.
(b) Construct a 4 £ 4 matrix A and a vector of initial conditions X0
such that when you solve
Xdot= AX; X(0) = X0
 we get x1 = te^(-3t)*sin7t. Also calculate x2, x3 and x4 for your
choice of A and X0.