[SOLVED] 18.06 Unit 2 Exam

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 b_i = 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 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 a₄ is a new vector and a₁,a₂,a₃,a₄ are independent. Which of these (if any) is the new Gram-Schmidt vector q₄ ? (P_A and P_Q from above)

1. 2.                           3.

kP_Qa₄k                    k norm of that vector k                     ka₄ − P_Aa₄k

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