MAT3701 Assignment 4 2026 - Due 4 September 2026

MAT3701 ASSIGNMENT 2026
DUE 4 SEPTEMBER 2026
3


Question 1

Let x = (2, 1 + i, i) and y = (2 − i, 2, 1 + 2i) be vectors in C3 . Use the standard inner product on C .

(1.1) Calculate

(1.1.1) ⟨x, y⟩.

(1.1.2) ∥x∥.

(1.1.3) ∥y∥.

(1.2) Confirm that the Cauchy-Schwarz inequality holds for these vectors.




Question 1

Topic: Standard Inner Product on ℂⁿ, Norm, and Cauchy-Schwarz Inequality

Friedberg, Insel & Spence, Linear Algebra, 4th Edition, Section 6.1: "Inner Products and Norms" (pp. 329-340)


(1.1.1) Calculate ⟨x, y⟩

⟨x, y⟩ = 8 + 5i



The standard inner product on ℂⁿ is defined as:

⟨x, y⟩ = x1 y1 + x2 y2 + ⋯ + xn yn
​ ​ ​ ​ ​ ​ ​ ​ ​




As stated in Friedberg Section 6.1 (p. 329), "If F = C , this gives the standard inner product on Cn ". This is also
confirmed in MAT3701 materials where the standard inner product on ℂⁿ is defined as ⟨x, y⟩ = y ∗ x for column
vectors.

, (1.1.2) Calculate ‖x‖



∥x∥ = 7

The norm (or length) of a vector is defined as:

∥x∥ = ⟨x, x⟩ = ∣x1 ∣2 + ∣x2 ∣2 + ⋯ + ∣xn ∣2

From Friedberg Section 6.1 (p. 330), the norm is derived from the inner product. The computation follows from the
property that ⟨x, x⟩ = ∣x1 ∣2 + ⋯ + ∣x 2
n ∣ .




(1.1.3) Calculate ‖y‖

∥y∥ = 14



Same as (1.1.2) above. By the definition of norm in Section 6.1.




(1.2) Confirm Cauchy-Schwarz Inequality



∣⟨x, y⟩∣ = 89 ≤ 98 = ∥x∥ ⋅ ∥y∥



The Cauchy-Schwarz Inequality (Friedberg, Section 6.1, Theorem 6.2, p. 331) states:

∣⟨x, y⟩∣ ≤ ∥x∥ ⋅ ∥y∥

This is a fundamental property of any inner product space. Equality holds if and only if one vector is a scalar multiple
of the other.