MAT3701 Assignment 4 Solutions 2026
UNISA
REMEMBER THE INSTRUCTIONS SAYS HAND WRITE !!!!!!!!!!!!!!!!!!!!!!!
, QUESTION 1
Given
𝑥 = (2, 1, +𝑖𝑖)
and
𝑦 = (2, −, 𝑖21 + 2𝑖).
We use the standard inner product on ℂ3 .
For complex vectors,
⟨𝑥 , 𝑦⟩ = 𝑥‾1 𝑦1 + 𝑥‾2 𝑦2 + 𝑥‾3 𝑦3 .
1.1.1 Calculate ⟨𝒙, 𝒚⟩.
First find the complex conjugate of 𝑥:
𝑥 = (2, 1, +𝑖𝑖)
therefore
𝑥‾ = (2, 1, −𝑖 − 𝑖).
Hence,
⟨𝑥 , 𝑦⟩ = (2)(2 − 𝑖) + (1 − 𝑖)(2) + (−𝑖)(1 + 2𝑖).
Calculate each term:
(2)(2 − 𝑖) = 4 − 2𝑖,
(1 − 𝑖)(2) = 2 − 2𝑖,
and
(−𝑖)(1 + 2𝑖) = −𝑖 − 2𝑖 2
= −𝑖 + 2
= 2 − 𝑖.
Therefore,
⟨𝑥, 𝑦⟩ = (4 − 2𝑖) + (2 − 2𝑖) + (2 − 𝑖)
= 8 − 5𝑖.
UNISA
REMEMBER THE INSTRUCTIONS SAYS HAND WRITE !!!!!!!!!!!!!!!!!!!!!!!
, QUESTION 1
Given
𝑥 = (2, 1, +𝑖𝑖)
and
𝑦 = (2, −, 𝑖21 + 2𝑖).
We use the standard inner product on ℂ3 .
For complex vectors,
⟨𝑥 , 𝑦⟩ = 𝑥‾1 𝑦1 + 𝑥‾2 𝑦2 + 𝑥‾3 𝑦3 .
1.1.1 Calculate ⟨𝒙, 𝒚⟩.
First find the complex conjugate of 𝑥:
𝑥 = (2, 1, +𝑖𝑖)
therefore
𝑥‾ = (2, 1, −𝑖 − 𝑖).
Hence,
⟨𝑥 , 𝑦⟩ = (2)(2 − 𝑖) + (1 − 𝑖)(2) + (−𝑖)(1 + 2𝑖).
Calculate each term:
(2)(2 − 𝑖) = 4 − 2𝑖,
(1 − 𝑖)(2) = 2 − 2𝑖,
and
(−𝑖)(1 + 2𝑖) = −𝑖 − 2𝑖 2
= −𝑖 + 2
= 2 − 𝑖.
Therefore,
⟨𝑥, 𝑦⟩ = (4 − 2𝑖) + (2 − 2𝑖) + (2 − 𝑖)
= 8 − 5𝑖.
