linear algebra review

Math 480: Matrix Theory and Applied Linear Algebra - Midterm 2

Thursday, November 12, 2015 Time: 10:30 - 11:30 AM (In SH 114)

Name: ————————————————————

Instructions: Be sure you have a complete exam. Show your work; little or no credit will be given for unsupported answers. Finally, before you start to work a problem, be sure that you understand what is being asked and that you are following instructions.

Question One: (? points) Find a basis for the orthogonal complement of V =

   

a b c d

  : a − b = d

 .

Question Two: (? points) Find a vector that is orthogonal to both

  11

0

  and

  11

1

 .

Question Three: (? points) What is the dimension of the orthogonal complement of span

    12

0

  ,

  24

0

   ?

Question Four: (? points) Let W = span

   

0 1 0 1

  ,

 

0 1 1 1

   .

(a) Find an orthonormal basis for W .

(b) What is the orthogonal projection of

 

1 2 1 0

  onto W .

(c) Write

 

1 0 0 0

  as the sum of a vector in W and a vector in W⊥.

(d) Find the projection matrix P corresponding to orthogonal projection onto W .

Question Five: (? points) If A and B are 3 × 3 matrices with det(A) = 4 and det(B) = 1. What is the determinant of C = 2AT A−1BA?

Question Six: (? points) Let A be a n × n matrix with AT = A−1. What can you say about det(A)?

Question Seven: (? points) Use Cramer’s rule to solve the system:

x1 + x2 + x3 = 4 x1 − x2 − x3 = 0 x1 + 2x2 + 3x3 = 9

1