# [Solved] 18.06 Unit 2 Exam

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Suppose q1,q2,q3 are orthonormal vectors in R3 . Find allpossiblevalues

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.

• 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 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.

• Give formulas using Q and A for the projection matrices PQ and PA onto the column spaces of Q and A.
• Is PQ = PA and why ? What is PQ times Q ? What is det PQ ?
• 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 ka4PAa4k

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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[Solved] 18.06 Unit 2 Exam
10 USD \$