Suppose q_{1},q_{2},q_{3} are orthonormal vectors in R^{3} . Find allpossiblevalues
for these 3 by 3 determinants and explain your thinking in 1 sentence each.
(c) det q_{1} q_{2} q_{3} times det q_{2} q_{3} q_{1} =
2
Suppose we take measurements at the 21 equally spaced times t = −10, −9,…, 9, 10.
All measurements are b_{i} = 0 except that b_{11} = 1 at the middle time t = 0.
 Using least squares, what are the best C and D to fit those 21 points by a straight line C + Dt?
 You are projecting the vector b onto what subspace? (Give a ) Find a nonzero vector perpendicular to that subspace.
4
The GramSchmidt method produces orthonormal vectors q_{1},q_{2},q_{3}
from independent vectors a_{1},a_{2},a_{3} in R^{5} . Put those vectors into the columns of 5 by 3 matrices Q and A.
 Give formulas using Q and A for the projection matrices P_{Q} and P_{A} onto the column spaces of Q and A.
 Is P_{Q} = P_{A} and why ? What is P_{Q} times Q ? What is det P_{Q} ?
 Suppose a_{4} is a new vector and a_{1},a_{2},a_{3},a_{4} are independent. Which of these (if any) is the new GramSchmidt vector q_{4} ? (P_{A} and P_{Q} from above)
 2. 3.
kP_{Q}a_{4}k k norm of that vector k ka_{4} − P_{A}a_{4}k
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
 The determinant of A is a polynomial in ×. What is the largest possible degree of that polynomial? Explain your
 If those 9 numbers give the identity matrix I, what is det A? Which values of × give det A = 0 ?
× × × ×
×_{ }0
A =
× 0 × 1
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