MAT 242 Dr. Ibrahim
SLN: 80071 Fall, 2007
MAT 242 Assignment D
Assignment due on Thursday, Nov. 8 in class.
D1: Prove that the system of linear equations
x+2y= 4
x+y= 5
3x+5y= 12
is inconsistent and find a least squares solution of the system.
D2: The points (0,0), (1,0), (2,1), (3,4), and (4,8) are required to lie on a parabola
y=a+bx +cx2. Find a least squares solution for a, b, c. Also prove that no parabola passes
through these points.
D3: Find the orthogonal projection of the vector b=
1
−1
1
onto the line given by the
vector a=
1
2
−2
. Give the orthogonal projection of any vector X=
x
y
z
onto the same
line.
D4: A subspace Shas basis
{a=
2
−1
0
1
, b =
5
0
−1
2
, c =
0
5
4
−1
}
a) What are the dot products aTb,aTc,bTc? Are the basis vectors orthogonal?
Now lets compute a new basis ˆa, ˆ
b, ˆcfor the same subspace. Start by letting ˆa=a.
b) Compute the projection P b of bonto the line described by a. What is the error (b−P b)?
Call this error vector ˆ
b.
c) Compute the projection P1cof conto the plane described by aand b. What is the
error (c−P1c)? Call this error vector ˆc. Does ˆcchange if we project onto the plane
with basis ˆaand ˆ
binstead? Why or why not?
d) What are the dot products ˆaTˆ
b, ˆaTˆc,ˆ
bTˆc? Are the new basis vectors orthogonal?
Now normalize the vectors to unit length. This is the Gram-Schmidt Process.
