In Exercises 3-determine whether each set equipped with the given operations is a vector space. Forvector spaces identify the vector space axioms that fail.3. The set of all real numbers with the standard operations of addition and multiplication.Answer:The set is a vector space with the given operations.

We have to prove set of all real number is vector space under standard operation. Let R be the set…

Answered: 3. The set of all real numbers with the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let W be the set of all vectors of the form shown on the right, where a, b, and c represent arbitrary real numbers. Find a set S of vectors that spans W or give an example or an explanation to show that W is not a vector space. 4a + 2b 5b + 2c 8c + 8a 7b Select the correct choice and fill in the answer box as needed to complete your choice. O A. A spanning set is S={} (Use a comma to separate answers as needed.) O B. There is no spanning set of W because W does not contain the zero vector. O C. There is no spanning set of W because W is not closed under vector addition. O D. There is no spanning set of W because W is not closed under scalar multiplication.
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Consider the set V of all real triples of the form (a, b, c) where c = a – 2b under the usual vector addition and scalar multiplication of R3 . Give the following (fill in the blanks) examples of 3-tuples in V as requested. Write your answers as comma-delimited triples using parentheses. For ease in reading, place a space after each comma. If no such requested element exists, write the word "none" in the box (no quotation marks). a. a specific non-zero element of V whose first and third components are the same: b. a specific element of V whose third component is 6. c. a non-zero specific element of V whose second component is -1. d. a specific element of R3 that is not in V. e. a specific element of v in V for which -v is not in V. please only answer d and e

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3. The set of all real numbers with the standard operations of addition and multiplication. Answer: The set is a vector space with the given operations.