MAT3701ASSIGNMENT42026
DUE4SEPTEMBER 2026
3
Question 1
Let x = (2, 1 + i, i) and y = (2 - i. 2, 1 + 2i) he vectors in C3. Use the standard inner product on C .
(1.1) Calculate
(1.1.1) (x,y).
( 1. 1.2) llxll.
(1.1.3) IIYII-
(1.2) Confirm that the Cauchy-Schwarz inequality holds for these vectors.
Question 1
Topic: Standard Inner Product on en, Norm, and Cauchy-Schwarz Inequality
Friedberg, lnsel & 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 en is defined as:
(x,y) = xu1 1 + x2y2 + ••• + x,,y,,
As stated in Friedberg Section 6.1 (p. 329), "If F = C, this gives the standard inner product on C" ". This is also
confirmed in MAT3701 materials where the standard inner product on e n is defined as (x,y) = y*x for column
vectors.
, (1.1.2) Calculate llxll
llxll=V7
The norm (or length) of a vector is defined as:
llxll = J(x,x) = J1x112 + lx212 + ·•· + lx,,12
From Friedberg Section 6.1 (p. 330), the norm is derived from the inner product. The computation follows from the
2
1
property that (x,x) = lx11 + • • • + I;,
(1.1.3) Calculate IIYII
llyll=Vl4
Same as (1.1.2) above. By the definition of norm in Section 6.1.
(1.2) Confirm Cauchy-Schwarz Inequality
l(x,y)I = v'89 :S v'98 = llxll • llyll
The Cauchy-Schwarz Inequality (Friedberg, Section 6.1, Theorem 6.2, p. 331) states:
l(x,y)I :S llxll • llyll
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.
DUE4SEPTEMBER 2026
3
Question 1
Let x = (2, 1 + i, i) and y = (2 - i. 2, 1 + 2i) he vectors in C3. Use the standard inner product on C .
(1.1) Calculate
(1.1.1) (x,y).
( 1. 1.2) llxll.
(1.1.3) IIYII-
(1.2) Confirm that the Cauchy-Schwarz inequality holds for these vectors.
Question 1
Topic: Standard Inner Product on en, Norm, and Cauchy-Schwarz Inequality
Friedberg, lnsel & 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 en is defined as:
(x,y) = xu1 1 + x2y2 + ••• + x,,y,,
As stated in Friedberg Section 6.1 (p. 329), "If F = C, this gives the standard inner product on C" ". This is also
confirmed in MAT3701 materials where the standard inner product on e n is defined as (x,y) = y*x for column
vectors.
, (1.1.2) Calculate llxll
llxll=V7
The norm (or length) of a vector is defined as:
llxll = J(x,x) = J1x112 + lx212 + ·•· + lx,,12
From Friedberg Section 6.1 (p. 330), the norm is derived from the inner product. The computation follows from the
2
1
property that (x,x) = lx11 + • • • + I;,
(1.1.3) Calculate IIYII
llyll=Vl4
Same as (1.1.2) above. By the definition of norm in Section 6.1.
(1.2) Confirm Cauchy-Schwarz Inequality
l(x,y)I = v'89 :S v'98 = llxll • llyll
The Cauchy-Schwarz Inequality (Friedberg, Section 6.1, Theorem 6.2, p. 331) states:
l(x,y)I :S llxll • llyll
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.
