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5. Which of the following sets of vectors in P2 is linearly dependent? A. {1,x, x²} B. {1+x,1 – x,2} C. {x² + 1, a2 – 1, 2a²} D. {1}

5. Which of the following sets of vectors in P2 is linearly dependent? A. {1,x, x²} B. {1+x,1 – x,2} C. {x² + 1, a2 – 1, 2a²} D. {1}

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Question 5: Identifying Linearly Dependent Sets of Vectors in \( P_2 \) **Question**: Which of the following sets of vectors in \( P_2 \) is linearly dependent? **Options**: - **A**. \(\{1, x, x^2\}\) - **B**. \(\{1 + x, 1 - x, 2\}\) - **C**. \(\{x^2 + 1, x^2 - 1, 2x^2\}\) - **D**. \(\{1\}\) **Explanation of Options**: - **Option A**: \(\{1, x, x^2\}\) - This set contains the polynomial basis vectors for the space \( P_2 \), which is a vector space of all polynomials of degree at most 2. This set is linearly independent by definition. - **Option B**: \(\{1 + x, 1 - x, 2\}\) - This set includes vectors that can form a linear combination potentially leading to some redundancy. - **Option C**: \(\{x^2 + 1, x^2 - 1, 2x^2\}\) - This set contains polynomials including different linear transformations of \( x^2 \). - **Option D**: \(\{1\}\) - This set consists of a single vector, which is implicitly linearly independent (a single vector in a vector space cannot be dependent). **Evaluation**: To determine which set is linearly dependent, we should check if any of the sets can form a linear combination where \( c_1v_1 + c_2v_2 + c_3v_3 = 0 \) (where \( c_i \) are scalar constants and not all zero). Answer: Set **B** consists of vectors whose combination leads to linear dependency because the vector \( 2 \) can be formed by linearly combining \( 1 + x \) and \( 1 - x \). Thus, the correct answer is: - **B**. \(\{1 + x, 1 - x, 2\}\)
Transcribed Image Text:### Question 5: Identifying Linearly Dependent Sets of Vectors in \( P_2 \) **Question**: Which of the following sets of vectors in \( P_2 \) is linearly dependent? **Options**: - **A**. \(\{1, x, x^2\}\) - **B**. \(\{1 + x, 1 - x, 2\}\) - **C**. \(\{x^2 + 1, x^2 - 1, 2x^2\}\) - **D**. \(\{1\}\) **Explanation of Options**: - **Option A**: \(\{1, x, x^2\}\) - This set contains the polynomial basis vectors for the space \( P_2 \), which is a vector space of all polynomials of degree at most 2. This set is linearly independent by definition. - **Option B**: \(\{1 + x, 1 - x, 2\}\) - This set includes vectors that can form a linear combination potentially leading to some redundancy. - **Option C**: \(\{x^2 + 1, x^2 - 1, 2x^2\}\) - This set contains polynomials including different linear transformations of \( x^2 \). - **Option D**: \(\{1\}\) - This set consists of a single vector, which is implicitly linearly independent (a single vector in a vector space cannot be dependent). **Evaluation**: To determine which set is linearly dependent, we should check if any of the sets can form a linear combination where \( c_1v_1 + c_2v_2 + c_3v_3 = 0 \) (where \( c_i \) are scalar constants and not all zero). Answer: Set **B** consists of vectors whose combination leads to linear dependency because the vector \( 2 \) can be formed by linearly combining \( 1 + x \) and \( 1 - x \). Thus, the correct answer is: - **B**. \(\{1 + x, 1 - x, 2\}\)
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