Back
0

Inner Product Spaces

Description: This quiz is designed to test your understanding of the fundamental concepts of inner product spaces, including the definition, properties, and applications of inner products.
Number of Questions: 15
Created by:
Tags: inner product spaces linear algebra mathematics
Start the Quiz
Attempted 0/15 Correct 0 Score 0
‹
›

In an inner product space, the inner product of two vectors (x) and (y) is denoted by:

  1. $\langle x, y \rangle$

  2. $\lVert x \rVert \cdot \lVert y \rVert$

  3. $\lVert x - y \rVert$

  4. $\lVert x + y \rVert$

Show answer
Correct Option: A
Explanation:

The inner product of two vectors (x) and (y) in an inner product space is denoted by (\langle x, y \rangle).

Which of the following properties is true for the inner product of two vectors (x) and (y) in an inner product space?

  1. Commutativity: (\langle x, y \rangle = \langle y, x \rangle)

  2. Associativity: (\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle)

  3. Distributivity: (\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle)

  4. All of the above

Show answer
Correct Option: D
Explanation:

In an inner product space, the inner product satisfies the properties of commutativity, associativity, and distributivity.

The norm of a vector (x) in an inner product space is defined as:

  1. $\lVert x \rVert = \sqrt{\langle x, x \rangle}$

  2. $\lVert x \rVert = \langle x, x \rangle$

  3. $\lVert x \rVert = \lVert x \rVert^2$

  4. $\lVert x \rVert = \lVert x - 0 \rVert$

Show answer
Correct Option: A
Explanation:

The norm of a vector (x) in an inner product space is defined as (\lVert x \rVert = \sqrt{\langle x, x \rangle}).

The Cauchy-Schwarz inequality in an inner product space states that:

  1. $\langle x, y \rangle^2 \le \lVert x \rVert^2 \lVert y \rVert^2$

  2. $\langle x, y \rangle^2 \ge \lVert x \rVert^2 \lVert y \rVert^2$

  3. $\langle x, y \rangle \le \lVert x \rVert \lVert y \rVert$

  4. $\langle x, y \rangle \ge \lVert x \rVert \lVert y \rVert$

Show answer
Correct Option: A
Explanation:

The Cauchy-Schwarz inequality in an inner product space states that (\langle x, y \rangle^2 \le \lVert x \rVert^2 \lVert y \rVert^2).

The parallelogram law in an inner product space states that:

  1. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 2\lVert x \rVert^2 + 2\lVert y \rVert^2$

  2. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = \lVert x \rVert^2 + \lVert y \rVert^2$

  3. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = \lVert x \rVert^2 - \lVert y \rVert^2$

  4. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 4\lVert x \rVert^2 + 4\lVert y \rVert^2$

Show answer
Correct Option: A
Explanation:

The parallelogram law in an inner product space states that (\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 2\lVert x \rVert^2 + 2\lVert y \rVert^2).

The Pythagorean theorem in an inner product space states that:

  1. $\lVert x + y \rVert^2 = \lVert x \rVert^2 + \lVert y \rVert^2$

  2. $\lVert x + y \rVert^2 = \lVert x \rVert^2 - \lVert y \rVert^2$

  3. $\lVert x + y \rVert^2 = 2\lVert x \rVert^2 + 2\lVert y \rVert^2$

  4. $\lVert x + y \rVert^2 = 4\lVert x \rVert^2 + 4\lVert y \rVert^2$

Show answer
Correct Option: A
Explanation:

The Pythagorean theorem in an inner product space states that (\lVert x + y \rVert^2 = \lVert x \rVert^2 + \lVert y \rVert^2).

In an inner product space, the angle between two vectors (x) and (y) is given by:

  1. $\theta = \arccos\left(\frac{\langle x, y \rangle}{\lVert x \rVert \lVert y \rVert}\right)$

  2. $\theta = \arcsin\left(\frac{\langle x, y \rangle}{\lVert x \rVert \lVert y \rVert}\right)$

  3. $\theta = \arctan\left(\frac{\langle x, y \rangle}{\lVert x \rVert \lVert y \rVert}\right)$

  4. $\theta = \arccot\left(\frac{\langle x, y \rangle}{\lVert x \rVert \lVert y \rVert}\right)$

Show answer
Correct Option: A
Explanation:

In an inner product space, the angle between two vectors (x) and (y) is given by (\theta = \arccos\left(\frac{\langle x, y \rangle}{\lVert x \rVert \lVert y \rVert}\right)).

The completeness of an inner product space means that:

  1. Every Cauchy sequence in the space converges to a limit in the space

  2. Every bounded sequence in the space converges to a limit in the space

  3. Every convergent sequence in the space is a Cauchy sequence

  4. Every divergent sequence in the space is a Cauchy sequence

Show answer
Correct Option: A
Explanation:

The completeness of an inner product space means that every Cauchy sequence in the space converges to a limit in the space.

Which of the following is an example of an inner product space?

  1. The set of all real numbers with the usual inner product

  2. The set of all complex numbers with the usual inner product

  3. The set of all polynomials with the usual inner product

  4. The set of all continuous functions on a closed interval with the usual inner product

Show answer
Correct Option:
Explanation:

All of the given examples are examples of inner product spaces.

The inner product space (\mathbb{R}^n) with the usual inner product is also known as:

  1. Euclidean space

  2. Hilbert space

  3. Banach space

  4. Fréchet space

Show answer
Correct Option: A
Explanation:

The inner product space (\mathbb{R}^n) with the usual inner product is also known as Euclidean space.

The inner product space (\mathbb{C}^n) with the usual inner product is also known as:

  1. Euclidean space

  2. Hilbert space

  3. Banach space

  4. Fréchet space

Show answer
Correct Option: B
Explanation:

The inner product space (\mathbb{C}^n) with the usual inner product is also known as Hilbert space.

The inner product space of all square-integrable functions on a closed interval with the usual inner product is also known as:

  1. Euclidean space

  2. Hilbert space

  3. Banach space

  4. Fréchet space

Show answer
Correct Option: B
Explanation:

The inner product space of all square-integrable functions on a closed interval with the usual inner product is also known as Hilbert space.

The inner product space of all continuous functions on a closed interval with the usual inner product is also known as:

  1. Euclidean space

  2. Hilbert space

  3. Banach space

  4. Fréchet space

Show answer
Correct Option: C
Explanation:

The inner product space of all continuous functions on a closed interval with the usual inner product is also known as Banach space.

The inner product space of all smooth functions on a closed interval with the usual inner product is also known as:

  1. Euclidean space

  2. Hilbert space

  3. Banach space

  4. Fréchet space

Show answer
Correct Option: D
Explanation:

The inner product space of all smooth functions on a closed interval with the usual inner product is also known as Fréchet space.

Inner product spaces are widely used in various fields of mathematics and its applications, including:

  1. Linear algebra

  2. Functional analysis

  3. Geometry

  4. Mathematical physics

Show answer
Correct Option:
Explanation:

Inner product spaces are widely used in various fields of mathematics and its applications, including linear algebra, functional analysis, geometry, and mathematical physics.

- Hide questions
ADVERTISEMENT

Find more quizzes from top tags

general knowledge
3615 Quizzes
119858 Questions
 technology
307 Quizzes
36705 Questions
 sports
963 Quizzes
19148 Questions
 history
1187 Quizzes
13475 Questions
 physics
598 Quizzes
13441 Questions
 biology
1160 Quizzes
12174 Questions
 science & technology
551 Quizzes
11015 Questions
 chemistry
490 Quizzes
9892 Questions
 maths
518 Quizzes
9324 Questions
 softskills
0 Quizzes
7131 Questions