Assignment # 6 (Makeup # 2) CSE 330 (01). 

Assignment # 6 (Makeup # 2) CSE 330 (01). 
Instructions for preparing the solution script:
• Write your name, ID#, and Section number clearly in the very front page.
• Write all answers sequentially.
• Start answering a question (not the part of the question) from the top of a new page.
• Write legibly and in orderly fashion maintaining all mathematical norms and rules. Prepare a single solution file.
• Start working right away. There is no late submission form. If you miss the deadline, you need to use the make-up
assignment to cover up the marks.
1. Answer the following questions:
(a) (3 marks) Show that the set,
S =
{
1
√
5
(
2, −1, 0
)T
,
1
√
30
(
1, 2, −5
)T
,
1
√
24
(
2, 4, 2
)T
}
is orthonormal.
(b) (6 marks) Consider the values of f(x) = sin x at the points x0 = 4, x1 = 9 and x2 = −6. Consider only up to 3
decimal places after rounding. Now, evaluate the best fit straight line using the Discrete Square Approximation
for the given function.
2. Consider a set of four data points given below:
f(0) = 3, f(4) = −2, f(−1) = 2, and f(1) = 1 .
Use the above data values to find the best fit polynomial of degree 2 by using the QR-decomposition method by
answering the questions below step by step.
(a) (2.5+1.5 marks) Identify the matrix A and b. Also identify the linearly independent column vectors u1, u2
and u3 from the matrix A. Explain why there are only three linearly independent vectors u1, u2 and u3.
(b) (1+2+3 marks) Using Gram-Schmidt process construct the orthonormal column matrices (or vectors) q1, q2
and q3 from the linearly independent column vectors obtained in the previous part, and then write down the
Q matrix.
(c) (3 marks) Now calculate the matrix elements of R , and write down the matrix R.
(d) (4 marks) Compute Rx and QTb.
(e) (4 marks) Let x = (a0 a1 a2)
T being the coefficients of the polynomial p2(x). Evaluate these coefficients and
write down the polynomial p2(x).
Motto: Mathematics is NOT difficult, but what is difficult is to believe that mathematics is NOT difficult.