Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Suppose V is a vector space with scalar fleld R. Thomas and Valentina want to prove that some set S C V is a subspace of V. Thomas decides to play it safe and use the Subspace Theorem, as stated in the notes, and check that the following three statements are true: (A) S contains the zero vector 0. (B) S is closed under vector addition. (C) Sis closed under scalar multiplication. But Valentina wants change condition (A) to something that they think is simpler. Valentina also thinks that (C) is unnecessary. As a result, Valentina proposes using just the following two conditions to prove that S is a subspace of V: (1) S {}. (2) If u, v € S then u + v € S. (a) Is it correct to claim that statement (A) can be replaced with statement (1) if (B) and (C) are kept? Select the option you believe to be true and then prove your assertion in the box tolow. O Yes, (A) can be replaced with (1). No, (A) cannot be replaced with (1). (The question continues below this essay box, make sure you scroll to the end of this question.)

Suppose V is a vector space with scalar fleld R. Thomas and Valentina want to prove that some set S C V is a subspace of V. Thomas decides to play it safe and use the Subspace Theorem, as stated in the notes, and check that the following three statements are true: (A) S contains the zero vector 0. (B) S is closed under vector addition. (C) Sis closed under scalar multiplication. But Valentina wants change condition (A) to something that they think is simpler. Valentina also thinks that (C) is unnecessary. As a result, Valentina proposes using just the following two conditions to prove that S is a subspace of V: (1) S {}. (2) If u, v € S then u + v € S. (a) Is it correct to claim that statement (A) can be replaced with statement (1) if (B) and (C) are kept? Select the option you believe to be true and then prove your assertion in the box tolow. O Yes, (A) can be replaced with (1). No, (A) cannot be replaced with (1). (The question continues below this essay box, make sure you scroll to the end of this question.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Suppose V is a vector space with scalar field R. Thomas and Valentina want to prove that some set SCV is a subspace of V. Thomas decides to play it safe and use the Subspace Theorem, as stated in the notes, and check that the following three statements are true; (A) S contains the zero vector 0. (B) S is closed under vector addition. (C) S is closed under scalar multiplication. But Valentina wants change condition (A) to something that they think is simpler. Valentina also thinks that (C) is unnecessary. As a result, Valentina proposes using just the following two conditions to prove that S is a subspace of V: (1) S±{} (2) If u, v E S then u +v€ S. (a) Is it correct to claim that statement (A) can be replaced with statement (1) if (B) and (C) are kept? Select the option you believe to be true and then prove your assertion in the box tolow. O Yes, (A) can be replaced with (1). O No, (A) cannot be replaced with (1). (The question continues below this essay box, make sure you scroll to the end of this question.) Eeuation Edtor A A- T BIUS X, x
Transcribed Image Text:Suppose V is a vector space with scalar field R. Thomas and Valentina want to prove that some set SCV is a subspace of V. Thomas decides to play it safe and use the Subspace Theorem, as stated in the notes, and check that the following three statements are true; (A) S contains the zero vector 0. (B) S is closed under vector addition. (C) S is closed under scalar multiplication. But Valentina wants change condition (A) to something that they think is simpler. Valentina also thinks that (C) is unnecessary. As a result, Valentina proposes using just the following two conditions to prove that S is a subspace of V: (1) S±{} (2) If u, v E S then u +v€ S. (a) Is it correct to claim that statement (A) can be replaced with statement (1) if (B) and (C) are kept? Select the option you believe to be true and then prove your assertion in the box tolow. O Yes, (A) can be replaced with (1). O No, (A) cannot be replaced with (1). (The question continues below this essay box, make sure you scroll to the end of this question.) Eeuation Edtor A A- T BIUS X, x
(b) Valentina says that using (2), it's easy to show that if u is in S then 2u, 3u, 4u, ...are also all in S and sa S is closed under scalar multuplication. The first part of Valentina's claim is equivalent to the following statement (Z) If u E Sand n is a postive integer then nu E S. In the box below, prove that (Z) follows from (2) but that this does not show that S is closed under scalar multiplication.
Transcribed Image Text:(b) Valentina says that using (2), it's easy to show that if u is in S then 2u, 3u, 4u, ...are also all in S and sa S is closed under scalar multuplication. The first part of Valentina's claim is equivalent to the following statement (Z) If u E Sand n is a postive integer then nu E S. In the box below, prove that (Z) follows from (2) but that this does not show that S is closed under scalar multiplication.
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Vector Space
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS