Explanation Given: W = { p ( x ) in P 2 : p ( 1 ) = p ( 3 ) } where P 2 is a polynomial of degree 2 or less. Approach: A subset A of a vector space V is called its subspace, if it satisfies the following properties: 1. The zero vector θ is in the subset A , 2. A is closed under addition and, 3. A is closed under scalar multiplication. Calculation: θ ( x ) = 0 x 2 + 0 x + 0 is a zero function for all x , in P 2 and θ ( 1 ) = θ ( 3 ) . θ ( 1 ) = ( 0 ( 1 ) 2 + 0 ( 1 ) + 0 ) = 0 + 0 = 0 θ ( 3 ) = ( 0 ( 3 ) 2 + 0 ( 3 ) + 0 ) = 0 + 0 = 0 Therefore, the zero vector θ is in the subset W . Let g ( x ) = x 2 − 4 x + 5 and h ( x ) = x 2 − 4 x + 7 be arbitrary functions of W . The sum of g ( x ) and h ( x ) is defined by s ( x ) s ( 1 ) = g ( 1 ) + h ( 1 ) = g ( 3 ) + h ( 3 ) = s ( 3 ) s ( 1 ) = 2 ( 1 ) 2 − 8 ( 1 ) + 12 = 2 − 8 + 12 = 6

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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Chapter 5.3, Problem 6E

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Whether the subset W={p(x) in P2: p(1)=p(3)} is a subspace of P2.

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Chapter 5.3, Problem 5E

Chapter 5.3, Problem 7E

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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - Prob. 4E Ch. 5.2 - Prob. 5E Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - Prob. 7E Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...

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Whether the subset W = { p ( x ) in P 2 : p ( 1 ) = p ( 3 ) } is a subspace of P 2 .

Solution Summary: The author explains that a subset of A is called its subspace if it satisfies the following properties.

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Whether the subset W={p(x) in P2: p(1)=p(3)} is a subspace of P2.

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