Find a basis for the subspace W p(x) E P2 p(5) = -p(-5)} of P2. A basis for W is { x2 -Ex: 1 , x
Find a basis for the subspace W p(x) E P2 p(5) = -p(-5)} of P2. A basis for W is { x2 -Ex: 1 , x
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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![**Find a Basis for the Subspace**
Given:
\[
W = \left\{ p(x) \in \mathcal{P}_2 \mid p(5) = -p(-5) \right\}
\]
of \(\mathcal{P}_2\).
**Solution:**
A basis for \(W\) is \(\{x^2 - 1, x\}\).
---
**Explanation of Notation and Concepts:**
- \(\mathcal{P}_2\) denotes the set of all polynomials of degree less than or equal to 2.
- \(W\) is described as the set of all polynomials \(p(x)\) in \(\mathcal{P}_2\) that satisfy the condition \(p(5) = -p(-5)\).
**Found Basis:**
- The basis \(\{x^2 - 1, x\}\) implies that any polynomial in \(W\) can be expressed as a linear combination of \(x^2 - 1\) and \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ad0bce7-1cf3-497a-a99e-802afa935450%2Fc8195ad8-a090-4c71-b979-a6f56a05f313%2Ft9k0uvm_processed.png&w=3840&q=75)
Transcribed Image Text:**Find a Basis for the Subspace**
Given:
\[
W = \left\{ p(x) \in \mathcal{P}_2 \mid p(5) = -p(-5) \right\}
\]
of \(\mathcal{P}_2\).
**Solution:**
A basis for \(W\) is \(\{x^2 - 1, x\}\).
---
**Explanation of Notation and Concepts:**
- \(\mathcal{P}_2\) denotes the set of all polynomials of degree less than or equal to 2.
- \(W\) is described as the set of all polynomials \(p(x)\) in \(\mathcal{P}_2\) that satisfy the condition \(p(5) = -p(-5)\).
**Found Basis:**
- The basis \(\{x^2 - 1, x\}\) implies that any polynomial in \(W\) can be expressed as a linear combination of \(x^2 - 1\) and \(x\).
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