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.