18.06SC Unit 2 Exam
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.
(c) det q1 q2 q3 times det q2 q3 q1 =
2
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
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)
1. 2. 3.
kPQa4k k norm of that vector k ka4 − PAa4k
6
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.
× × × ×
× 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 ?
× × × ×
× 0
A =
× 0 × 1
