Page 348 - 35Linear Algebra

348 Sample Final Exam

(d) Why is the angle between vectors V and W is not changed when you
replace them by PV and PW for P any orthogonal matrix?
(e) Explain how to choose an orthogonal matrix P such that MP = PD
where D is a diagonal matrix.
(f) For the choice of P above, define our final unknown vector Z by Y =
T
PZ. Find an expression for Y MY in terms of Z and the eigenvalues
of M.

z
(g) Call Z = . What equation do z and w obey? (Hint, write your
w
answer using λ, µ and g.)
(h) Central conics are circles, ellipses, hyperbolae or a pair of straight lines.
Give examples of values of (λ, µ, g) which produce each of these cases.

13. Let L: V → W be a linear transformation between finite-dimensional vector
spaces V and W, and let M be a matrix for L (with respect to some basis
for V and some basis for W). We know that L has an inverse if and only if
it is bijective, and we know a lot of ways to tell whether M has an inverse.
In fact, L has an inverse if and only if M has an inverse:

(a) Suppose that L is bijective (i.e., one-to-one and onto).
i. Show that dim V = rank L = dim W.
ii. Show that 0 is not an eigenvalue of M.
iii. Show that M is an invertible matrix.
(b) Now, suppose that M is an invertible matrix.
i. Show that 0 is not an eigenvalue of M.
ii. Show that L is injective.
iii. Show that L is surjective.

14. Captain Conundrum gives Queen Quandary a pair of newborn doves, male
and female for her birthday. After one year, this pair of doves breed and
produce a pair of dove eggs. One year later these eggs hatch yielding a new
pair of doves while the original pair of doves breed again and an additional
pair of eggs are laid. Captain Conundrum is very happy because now he will
never need to buy the Queen a present ever again!
Let us say that in year zero, the Queen has no doves. In year one she has
one pair of doves, in year two she has two pairs of doves etc... Call F n the
number of pairs of doves in years n. For example, F 0 = 0, F 1 = 1 and
F 2 = 1. Assume no doves die and that the same breeding pattern continues

348