Name:
CSU ID:
Homework 12
December 4, 2015
1. Replace the basis given below, with an orthonormal basis. (Essentially,
perform the QR decomposition.)




~v1 = 
−2
3
0
4



,





~v2 = 

4
−7
4
−6


,





~v3 = 
7
−1
2
8





2. Given the previous problem, find the solution of A~x = ~b where the
columns of A are ~v1 , ~v2 , ~v3 and ~b = [1, 1, 1, 1]T . What is the least
squared error (remember to divide by 4)?
1
3. (a) Given the inner product hu, vi = −1
u(x)v(x), find two nonzero
polynomials of degree less than or equal to 3 that are orthogonal
to u(x) = sin(x).
R
(b) Given the same inner product as in (a), find two nonzero polynomials of degree less than or equal to 3 that are orthogonal to
cos(x).
4. Consider the polynomials p~1 (x) = 1 − x + x2 , p~2 (x)
= 4 + x − 3x2 , and
R1
p~3 (x) = x. Assuming the inner product hf, gi = 0 f (x)g(x)dx replace
the vectors in the same way as the QR algorithm for vectors; i.e.,
(a) Replace p~1 (x) with a vector of length 1. Call it ~q1 (x).
(b) Project p~2 (x) onto ~q1 (x). That is, evaluate h~
p2 , ~q1 i and plug into
the projection definition.
(c) Find w
~ 2 (x), the vector orthogonal to ~q1 (x) such that span{~q1 (x), w
~ 2 (x)} =
~
span{~
p1 (x), p2 (x)}.
(d) Replace w
~ 2 (x) with a vector in the same direction but of length
1. Call the result q2 (x).
(e) Find w
~ 3 (x), the vector orthogonal to ~q1 (x), ~q2 (x) such that span{~q1 (x), ~q2 (x), w
~ 3 (x)} =
span{~
p1 (x), p~2 (x), p~3 (x)}
(f) Find ~q3 (x), the vector of length 1 in the same direction as w
~ 3 (x).