Department of Mathematics, IIT Madras
Linear Algebra for Engineers (MA2031)
Assignment-2 (Inner Product spaces)
1. Why is the map h , i not an inner product on the given vector space?
(a) h𝑥, 𝑦i = 𝑎𝑐 for 𝑥 = (𝑎, 𝑏), 𝑦 = (𝑐, 𝑑) in R2 .
∫1
(b) h𝑥, 𝑦i = 0 𝑥 0 (𝑡)𝑦0 (𝑡) 𝑑𝑡 for 𝑥, 𝑦 ∈ 𝐶 1 [0, 1].
Ans: For each of these, construct a vector 𝑣 so that h𝑣, 𝑣i = 0.
2. Let 𝐵 be a basis for a finite dimensional ips 𝑉 . Let 𝑦 ∈ 𝑉 be such that h𝑥, 𝑦i = 0 for all
𝑥 ∈ 𝐵. Show that 𝑦 = 0.
3. Let 𝑉 be an inner product space, and let 𝑥, 𝑦 ∈ 𝑉 . Show the following:
(a) k𝑥k ≥ 0.
(b) 𝑥 = 0 iff k𝑥k = 0.
(c) k𝛼𝑥k = |𝛼|k𝑥k, for all 𝛼 ∈ F.
(d) k𝑥 + 𝛼𝑦k = k𝑥 − 𝛼𝑦k for all 𝛼 ∈ F iff h𝑥, 𝑦i = 0.
(e) If k𝑥 + 𝑦k = k𝑥k + k𝑦k, then at least one of 𝑥, 𝑦 is a scalar multiple of the other.
4. Let 𝑉 be a complex ips. Show that Reh𝑖𝑥, 𝑦i = −Imh𝑥, 𝑦i for all 𝑥, 𝑦 ∈ 𝑉 .
5. (Polarization Identity): Let 𝑉 be an ips over F. Let 𝑥, 𝑦 ∈ 𝑉 . Show the following:
(a) If F = R, then 4h𝑥, 𝑦i = k𝑥 + 𝑦k 2 − k𝑥 − 𝑦k 2 .
(b) If F = C, then 4h𝑥, 𝑦i = k𝑥 + 𝑦k 2 − k𝑥 − 𝑦k 2 + 𝑖k𝑥 + 𝑖𝑦k 2 − 𝑖k𝑥 − 𝑖𝑦k 2 .
6. For 1 ≤ 𝑗 ≤ 𝑛, let 𝑎 𝑗 ≥ 0 and 𝑏 𝑗 ≥ 0. Show that
𝑛
Õ 2 𝑛
Õ 𝑛
Õ 2
𝑎𝑗𝑏𝑗 ≤ ( 𝑗 𝑎 𝑗 )2 𝑎𝑗/𝑗 .
𝑗=1 𝑗=1 𝑗=1
7. Let 𝑊 = {𝑥 ∈ R4 : 𝑥 ⊥ (1, 0, −1, 1), 𝑥 ⊥ (2, 3, −1, 2)}, where R4 is the real ips with standard
inner product. Show that 𝑊 is a subspace of R4 . Also, find a basis for 𝑊.
8. Let {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } be an orthogonal set in a real ips 𝑉 . Let 𝑎 1 , ..., 𝑎 𝑘 ∈ R. Is it true that
𝑎𝑖 𝑢𝑖 k 2 = 𝑖=1 |𝑎𝑖 | 2 k𝑢𝑖 k 2 ? Ans: Yes.
Í𝑘 Í𝑘
k 𝑖=1
9. Show that {sin 𝑡, sin(2𝑡), . . . , sin(𝑚𝑡)} is linearly independent in 𝐶 [0, 2𝜋].
10. Consider R3 with the standard inner product. Apply Gram-Schmidt process on the given
set of vectors.
(a) {(1, 2, 0), (2, 1, 0), (1, 1, 1)} (b) {(1, 1, 1), (1, −1, 1), (1, 1, −1)}
Ans: (a) (1, 2, 0), (6/5, −3/5, 0), (0, 0, 1) (b) (1, 1, 1), (2, −4, 2)/3, (1, 0, −1)
11. Consider R4 with the standard inner product. In each of the following, find the set of all
vectors orthogonal to both 𝑢 and 𝑣.
(a) 𝑢 = (1, 2, 0, 1), 𝑣 = (2, 1, 0, −1) (b) 𝑢 = (1, 1, 1, 0), 𝑣 = (1, −1, 1, 1)
Ans: (a) span {(−1, 1, 0, 1), (0, 0, 1, 0)} (b) span {(−6, 1, 5, 2), (0, 1, −1, 1)}.
12. Consider the polynomials 𝑢 0 (𝑡) = 1, 𝑢 1 (𝑡) = 𝑡, 𝑢 2 (𝑡) = 𝑡 2 in R2 [𝑡]. Using Gram-Schmidt
orthogonalization, find orthogonal polynomials obtained from 𝑢 1 , 𝑢 2 , 𝑢 3 with respect to
the following inner products:
1
∫1 ∫1
(a) h𝑝, 𝑞i = 0
𝑝(𝑡)𝑞(𝑡) 𝑑𝑡 (b) h𝑝, 𝑞i = −1
𝑝(𝑡)𝑞(𝑡) 𝑑𝑡
Ans: (a) 1, 𝑡 − 1/2, 𝑡 2 − 𝑡 + 1/6. (b) 1, 𝑡, 𝑡 2 − 1/3.
13. Find the best approximation of 𝑣 ∈ 𝑉 from 𝑈 in the following:
(a) 𝑉 = R3 , 𝑣 = (1, 2, 1), 𝑈 = span {(3, 1, 2), (1, 0, 1)}.
∫1
(b) 𝑉 = R3 [𝑡], 𝑣 = 𝑡 3 , 𝑈 = span {1, 1 + 𝑡, 1 + 𝑡 2 }, h𝑝, 𝑞i = 0 𝑝(𝑡)𝑞(𝑡) 𝑑𝑡.
Ans: (a) (5/3, 4/3, 1/3) (b) −19/20 − 3𝑡/5 + 3𝑡 2 /2
