Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 6 – Thursday, September 23, 1999
NAMES
Hand quiz 6 in at the end of the class period. Be sure to put all your names on the paper.
I) In this problem, .you are to go through the process of …nding the line which best …ts the
data points (¡3; 2); (¡1; ¡1); and (2; 3). Let’s agree to call the line y = ®x + ¯
Give the system of equations which results when the points are put into the equation, and
then put the system into matrix form Ax = b:
Give the geometric picture of A (include N(A); R(A); and b):
Go through the steps leading to the normal equations (you must include x and p in the above
sketch and explain what they are).
Solve the normal equation (you can use Maple here).
What is the line that best …ts the data?
II) Find the parabola which best …ts the data points (¡1; 3); (0; 2); (2; ¡1); (3; 1); (5; 4): I
want to see the model, the system of equations (as equations and also in matrix form), a
rough sketch of the geometric picture of the matrix, the normal equations, and the parabola.
Note that I am not asking you to derive the normal equations.
III) Finding the projection matrix. Suppose that M is a k–dimensional subspace (line or
plane or...) of Rn : Here’s how to …nd the matrix that projects vectors onto M . First …nd
basis for M; say v1 ; v2 ; vk : Note that each of these vectors will have n components. Let A
be the matrix whose columns are the v’s. Thus A is a n £ k matrix and
³
´¡1
P = A A> A
A>
a) By putting dimensions under the matrices, show that P is n £ n:
b) Now for a concrete example. Let M be the plane 2x + 3y ¡ z = 0 in R3 :
A basis for M is
Form the matrix A:
Form the projection matrix P (Use Maple here).
0
1 0
1
5
1
C
C B
Thus, if x 2 R3 ; then P x is the projection of x onto M: Try it with the vectors B
@ 1 A;@ 3 A;
2
1
1
1 0
0
4
1
B
C B
C
@ 0 A;@ 6 A:
2
¡2
Explain why the answers to the last two vectors are true.