Description
1 (24 pts.) Suppose q1, q2, q3 are orthonormal vectors in R3. Find all possible values
for these 3 by 3 determinants and explain your thinking in 1 sentence each.
(a) det q1 q2 q3 =
(b) det q1 + q2 q2 + q3 q3 + q1 =
(c) det q1 q2 q3 times det q2 q3 q1 =
2
2 (24 pts.) Suppose we take measurements at the 21 equally spaced times t = −10, −9, . . . , 9, 10.
All measurements are bi = 0 except that b11 = 1 at the middle time t = 0.
(a) Using least squares, what are the best C� and D� to fit those 21 points
by a straight line C + Dt ?
(b) You are projecting the vector b onto what subspace ? (Give a basis.)
Find a nonzero vector perpendicular to that subspace.
4
3 (9 + 12 + 9 pts.) The Gram-Schmidt method produces orthonormal vectors q1, q2, q3
from independent vectors a1, a2, a3 in R5. Put those vectors into the columns
of 5 by 3 matrices Q and A.
(a) Give formulas using Q and A for the projection matrices PQ and PA
onto the column spaces of Q and A.
(b) Is PQ = PA and why ? What is PQ times Q ? What is det PQ ?
(c) Suppose a4 is a new vector and a1, a2, a3, a4 are independent. Which of
these (if any) is the new Gram-Schmidt vector q4 ? (PA and PQ from
above)
T
4
T
3
T
4
T
2
a a2 a a3
a a2 a a3
kPQa4k k norm of that vector k ka4 − PAa4k
Ta4 a1
PQa4
a4 − a1 − a2 − a3 a4 − PAa4 T
1a a1
1. 2. 3.
6
4 (22 pts.) Suppose a 4 by 4 matrix has the same entry × throughout its first row and
column. The other 9 numbers could be anything like 1, 5, 7, 2, 3, 99, π, e, 4.
× × × ×
A =
× any numbers
× any numbers
× any numbers
(a) The determinant of A is a polynomial in ×. What is the largest
possible degree of that polynomial ? Explain your answer.
(b) If those 9 numbers give the identity matrix I, what is det A ? Which
values of × give det A = 0 ?
× × × ×
× 1 0 0
A =
× 0 1 0
× 0 0 1
8
18.06SC Linear Algebra
