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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
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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