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{1+2x²,−1+x+x²,2+z=x²} (a) linearly independent (b) linearly dependent (c) spans P₂ (d) is a basis for P₂

{1+2x²,−1+x+x²,2+z=x²} (a) linearly independent (b) linearly dependent (c) spans P₂ (d) is a basis for P₂

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
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### Linear Algebra: Multiple Choice Questions on Polynomial Sets #### Question 3 Consider the set of polynomials: \(\{1 + 2x^2, -1 + x + x^2, 2 + x - x^2\}\). Options: - (a) Linearly independent - (b) Linearly dependent - (c) Spans \( P_2 \) - (d) Is a basis for \( P_2 \) #### Question 4 Consider the set of polynomials: \(\{1, 1 + x + x^2\}\). Options: - (a) Linearly independent - (b) Linearly dependent - (c) Spans \( P_2 \) - (d) Is a basis for \( P_2 \) In these questions, we assess the properties of given sets of polynomials in the context of their linear independence, dependence, spanning, and whether they form a basis for the polynomial space \(P_2\). **Explanation of Terms:** - **Linearly Independent**: A set of vectors (or polynomials) is linearly independent if no vector in the set is a linear combination of the others. - **Linearly Dependent**: A set of vectors is linearly dependent if at least one vector in the set can be written as a linear combination of the others. - **Spans \( P_2 \)**: A set of vectors (or polynomials) spans a space if any vector in that space can be written as a linear combination of the vectors in the set. - **Basis for \( P_2 \)**: A set of vectors that is both linearly independent and spans the space. For polynomials of degree up to 2, \(P_2\) is the space of all polynomials of degree 2 or less, and a basis for this space must consist of exactly three linearly independent polynomials.
Transcribed Image Text:### Linear Algebra: Multiple Choice Questions on Polynomial Sets #### Question 3 Consider the set of polynomials: \(\{1 + 2x^2, -1 + x + x^2, 2 + x - x^2\}\). Options: - (a) Linearly independent - (b) Linearly dependent - (c) Spans \( P_2 \) - (d) Is a basis for \( P_2 \) #### Question 4 Consider the set of polynomials: \(\{1, 1 + x + x^2\}\). Options: - (a) Linearly independent - (b) Linearly dependent - (c) Spans \( P_2 \) - (d) Is a basis for \( P_2 \) In these questions, we assess the properties of given sets of polynomials in the context of their linear independence, dependence, spanning, and whether they form a basis for the polynomial space \(P_2\). **Explanation of Terms:** - **Linearly Independent**: A set of vectors (or polynomials) is linearly independent if no vector in the set is a linear combination of the others. - **Linearly Dependent**: A set of vectors is linearly dependent if at least one vector in the set can be written as a linear combination of the others. - **Spans \( P_2 \)**: A set of vectors (or polynomials) spans a space if any vector in that space can be written as a linear combination of the vectors in the set. - **Basis for \( P_2 \)**: A set of vectors that is both linearly independent and spans the space. For polynomials of degree up to 2, \(P_2\) is the space of all polynomials of degree 2 or less, and a basis for this space must consist of exactly three linearly independent polynomials.
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