Page 99 - 35Linear Algebra

4.5 Review Problems 99

where the vector p labels a given point on the plane and n is a vector
101
normal to the plane. Let N and P be vectors in R and
 1 
x
 x 2 
X =  .  .


.
 . 
x 101
What kind of geometric object does N · X = N · P describe?

7. Let
   
1 1
1
   2 
   
1 3
   
u =   and v = 

. .
 .   . 
 .   . 
1 101
Find the projection of v onto u and the projection of u onto v. (Hint:
Remember that two vectors u and v define a plane, so first work out
how to project one vector onto another in a plane. The picture from
Section 14.4 could help.)

8. If the solution set to the equation A(x) = b is the set of vectors whose
2
2
tips lie on the paraboloid z = x + y , then what can you say about
the function A?
9. Find a system of equations whose solution set is

 
    
 1 −1 0 
 
1 −1 0
 
     
  + c 1   + c 2   c 1 , c 2 ∈ R .
 2  0   −1 
 
 

0 1 −3
 
Give a general procedure for going from a parametric description of a
hyperplane to a system of equations with that hyperplane as a solution
set.
10. If A is a linear operator and both v and cv (for any real number c) are
solutions to Ax = b, then what can you say about b?

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