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MATH2501 Linear Algebra

创建日期：2024-07-01 17:43:33

所属分类：数学mathematics

标签： MATH2501

Linear Algebra

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MATH2501 Linear Algebra
SESSION 2, 2020 TEST 2 Version A
Student’s Surname Initials Student Number
Questions: 4 Pages: 1 Total marks: 20 Time allowed: 60 minutes
Below, the term scalar product is synonym to the term inner product.
Q1 [5 marks] Let V be a vector space with scalar product V =
(
V, 〈·, ·〉).
a) Explain what it means that the function 〈·, ·〉 : V × V → R is positive-definite. Hint : Possible answer may
start as follows
The function 〈·, ·〉 : V × V → R is called positive-definite if [. . . ]
b) Give an example of a function 〈·, ·〉 : R2 ×R2 → R which is not positive-definite. Explain why your example
fails to be positive-definite.
Q2 [6 marks] Let V be a vector space with scalar product V =
(
V, 〈·, ·〉).
a) Let x ∈ V . Give definition the symbol ‖x‖. Hint : Possible answer may start as follows
For any vector x ∈ V , the length ‖x‖ is defined by [. . . ]
b) Let x ∈ V and x 6= 0. Prove that
‖y‖ = 1 where y ∈ V and y = 1‖x‖x.
Make sure you carefully reference every scalar product axiom you use in your proof. Make sure you do
not reference axioms you do not use.
Q3 [4 marks] Let V be a vector space with scalar product V =
(
V, 〈·, ·〉).
a) Carefully write the statement the Cauchy-Schwartz Inequality. Hint : Possible answer may start as
follows
Let (V 〈·, ·〉) be a vector space with scalar product. For any vectors x,y ∈ V , the Cauchy-Schwartz Inequality
states that [. . . ]
b) Let V = R2. Copy the following statement into your work and replace every appearance of [. . .] with correct
expressions:
The Cauchy-Schwartz Inequality in R2 states that, for every x = (x1, x2) ∈ R2 and for every y = (y1, y2) ∈ R2,
the following inequality holds ∣∣∣x1y1 + [. . .]∣∣∣ ≤√x21 + [. . .]× [. . .].
Q4 [5 marks] Let V be a vector space with scalar product V =
(
V, 〈·, ·〉) and let S ⊆ V be subset.
a) Give definition of the orthogonal complement S⊥. Hint : Possible answer may start as follows
Let (V 〈·, ·〉) be a vector space with scalar product and let S ⊆ V be a subset. The orthogonal complement is
defined by [. . . ]
b) Let x ∈ V . Prove that, if x ∈ S and x ∈ S⊥, then x = 0. Make sure you carefully reference every scalar
product axiom you use in your proof.

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