Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
{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
See similar textbooks
icon
Related questions
Question
### 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.
Expert Solution
steps

Step by step

Solved in 6 steps with 6 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS