MAT3701
ASSIGNMENT 4
Linear Algebra III
FULL
SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 11
, SOLUTIONS:
Question 1
(1.1.1) Calculate ⟨𝑥, 𝑦⟩
The standard inner product on ℂ3 is defined as ⟨(𝑎1, 𝑎2, 𝑎3), (𝑏1, 𝑏2 , 𝑏3 )⟩ = 𝑎1 𝑏‾1 + 𝑎2 𝑏‾2 +
𝑎3 𝑏‾3.
Given 𝑥 = (2, 1, +𝑖𝑖) and 𝑦 = (2, −, 𝑖21 + 2𝑖), we have:
‾ 𝑖) + (1 + 𝑖) ⋅ 2‾ + 𝑖 ⋅ (1 +‾ 2𝑖)
⟨𝑥, 𝑦⟩ = 2 ⋅ (2 −
= 2(2 + 𝑖) + (1 + 𝑖)(2) + 𝑖(1 − 2𝑖)
= (4 + 2𝑖) + (2 + 2𝑖) + (𝑖 − 2𝑖 2 ).
Since 𝑖 2 = −1, we get 𝑖 − 2𝑖 2 = 𝑖 − 2(−1) = 𝑖 + 2 = 2 + 𝑖.
Page 2 of 11
ASSIGNMENT 4
Linear Algebra III
FULL
SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 11
, SOLUTIONS:
Question 1
(1.1.1) Calculate ⟨𝑥, 𝑦⟩
The standard inner product on ℂ3 is defined as ⟨(𝑎1, 𝑎2, 𝑎3), (𝑏1, 𝑏2 , 𝑏3 )⟩ = 𝑎1 𝑏‾1 + 𝑎2 𝑏‾2 +
𝑎3 𝑏‾3.
Given 𝑥 = (2, 1, +𝑖𝑖) and 𝑦 = (2, −, 𝑖21 + 2𝑖), we have:
‾ 𝑖) + (1 + 𝑖) ⋅ 2‾ + 𝑖 ⋅ (1 +‾ 2𝑖)
⟨𝑥, 𝑦⟩ = 2 ⋅ (2 −
= 2(2 + 𝑖) + (1 + 𝑖)(2) + 𝑖(1 − 2𝑖)
= (4 + 2𝑖) + (2 + 2𝑖) + (𝑖 − 2𝑖 2 ).
Since 𝑖 2 = −1, we get 𝑖 − 2𝑖 2 = 𝑖 − 2(−1) = 𝑖 + 2 = 2 + 𝑖.
Page 2 of 11
