National Institute of Technology, Warangal
Department of MATHEMATICS
AY 2023-24 - March 2024 – Assignment - 03
Date: 12-03-2024 - Submit by: 18-03-2024 - 5 P.M
Program: B.Tech (Maths & Comp)
Class: I year II Semester
Course Code: MA 162
Title: COMPUTATIONAL LINEAR ALGEBRA
Topic: Orthogonality
1. Define inner product. Check whether following are the inner products for x, y ∈ V,
V = R2 , where x = (x1 , x2 ) , y = (y1 , y2 ).
(a) ⟨x, y⟩ = x1 x2 − y1 x2 − x1 y2 + 4y1 y2
(b) ⟨x, y⟩ = y1 (x1 + 2x2 ) + y2 (2x1 + 5x2 ).
2. Define inner product spaces. Let V be an inner product space having following property
⟨ax + by, cv + dw⟩ = ac⟨x, v⟩ + ad⟨x, w⟩ + bc⟨y, v⟩ + bd⟨y, w⟩.
Determine (a) ⟨x + y, x + y⟩ (b) ⟨x + y, x − y⟩.
3. Define norm function ∥ · ∥ on V . For (x, y) ∈ R2 , which of following are norms?
(a) ∥(x, y)∥ = (|x|2 + |y|2 )
1/2
(b) ∥(x, y)∥ = |x| + |y|
4. State the Cauchy-Schwarz inequality and verify that dot product in R2 satisfies the
inequality.
5. Compute the angle between vectors (x, y) and (−y, x) in R2 , x ̸= 0, y ̸= 0.
6. (a) Show that S = {(3, 1, 1), (−1, 2, 1), (−1/2, −2, 7/2)} is an orthogonal set using standard inner product.
(b) Let S be an orthogonal basis for R3 . Express (6, 1, −8) as a linear combination of
vectors in S.
7. Define orthonormal basis. Is {v1 , v2 , v3 } an orthonormal basis of R3 , where



v1 = 


√
3/ 11
√
1/ 11
√
1/ 11



,





v2 = 


√
−1/ 6
√
2/ 6
√
1/ 6



,





v3 = 


√
−1/ 66
√
−4/ 66
√
7/ 66



.


8. Prove Pythagorean theorem i.e., Suppose u, v ∈ V . If u, v are orthogonal, then
∥u + v∥2 = ∥u∥2 + ∥v∥2 .
9. Show that if ∥u + v∥ · ∥u − v∥ ≤ ∥u∥2 + ∥v∥2 .
10. Apply Gram Schmidt process to obtain
(a) {(−1, 0, 1), (1, −1, 0), (0, 0, 1)} in R3
(b) {1, t, t2 } of P2 with inner product
Z
1
⟨p, q⟩ =
p(t)q(t)dt.
0
where, P2 is the space of polynomials of degree at most 2.
11. Define orthogonal complement of a subset U of vector space V denoted by U ⊥ .
12. Define orthogonal projection. Then find the orthogonal projection of (−10, 2, 8) on
subspace W

 7
13. Let ȳ = 
6
spanned by (3, 12, −1).

 

 4 
 , ū =  
2
(a) Find orthogonal projection of ȳ onto ū
(b) Write ȳ as sum of two orthogonal vectors, one in space spanned by {ū} and one
orthogonal to {ū} (i.e., orthogonal decomposition).
14. Let V be the real inner product space consisting of the space of real valued continuous
functions on the interval, −1 ≤ t ≤ 1, with the inner product
Z 1
⟨f, g⟩ =
f (t)g(t)dt.
−1
Let W be the subspace of odd functions, i.e., f (−t) = −f (t). Find the orthogonal
complement of W .
2