1.
8 pts into HW
E = { 1, cost, cos2t , cos3t , cos4t , cos5t , cos6t } and
F = { 1, cost, cos2t, cos3t, cos4t, cos5t, cos6t}
be two ordered bases for the vector space of continuous functions.
Let
The following trigonometric identities will be helpful:
cos2t = -1 + 2 cos2t
cos3t = -3 cost + 4 cos3t
cos4t = 1 - 8 cos2t + 8 cos4t
cos5t = 5 cost - 20 cos3t + 16 cos5t
cos6t = -1 + 18 cos2t - 48 cos4t + 32 cos6t
(a)
Find [1]E, [cost]E , [cos2t]E , [cos3t]E , [cos4t]E , [cos5t]E , [cos6t]E .
(b)
Use part (a) to show that F is a basis for S, the subspace of functions
spanned by { 1, cost, cos2t , cos3t , cos4t , cos5t , cos6t }.
(c)
Find the transition matrix T from basis E to basis F. (you can use your
calculator)
(d)
Recall from calculus that integrals such as
(5 cos
3
t 6 cos 4 t + 5 cos 5 t 12 cos 6 t )dt
are tedious to compute. (you need to apply integration by parts repeatedly and
use the half angle formula). Instead, use the transition matrix T or T-1 (which
ever is appropriate) to compute the integral in an easier form.
2.
2 pts into HW
Crystal lattice for titanium has a hexagonal structure where the vectors
2.6
1.5 ,
0
0
3 and
0
0
0 in R3 form a basis for the unit cell. (the numbers given
4.8
here are in Angstrom units that are 10-8 cm. In alloys of titanium, some
additional atoms may be inside the unit cell.
(a)
1 / 2
One such site for additional atoms is 1 / 4 relative to the lattice basis.
1 / 6
Determine the coordinates if this site relative to the standard basis (and
understand while researchers prefer to use the lattice basis).
(b)
1 / 2
One other such site for additional atoms is 1 / 2 relative to the lattice basis.
1 / 3
Determine the coordinates if this site relative to the standard basis (and
understand while researchers prefer to use the lattice basis).
(c)
(for extra 2 points) For the artistically talented ones: Draw a 3-dimensional
figure of the hexagonal lattice structure and draw in those site (from parts a and b)
for additional atoms with colored pen.
3.
3 pts into HW
A scientist solves a nonhomogeneous system of ten linear equations in twelve
unknowns and finds that three of the unknowns are free variables. Can the
scientist be certain that, if the right side of the equations is changed, the new
nonhomogeneous system will have a solution. Explain your answer.