Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Transcription for Educational Website**
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**Problem: Understanding the Vector Space of Polynomials**
1. **Find the Dimension of \( P_3 \):**
What dimension is \( P_3 \) (the set of polynomials of degree 3 or less)?
[Input Box for Answer]
2. **Determine a Basis for \( P_3 \):**
Which of the following sets form a basis for \( P_3 \)?
- \(\{2x^2, x, 0, x^3\}\)
- \(\{2 + x^3, -2x^2, x, -5, 2x + 3\}\)
- \(\{3, -x^3, x\}\)
- \(\{1, 2x, -3x^2, 2x^3\}\)
- \(\{2x^2, x, -5, 2 + x^3\}\)
**Explanation:**
- **Dimension of \( P_3 \):** In linear algebra, the dimension of a vector space is the number of vectors in a basis for the space. For the space \( P_3 \), the basis is typically \(\{1, x, x^2, x^3\}\), indicating a dimension of 4.
- **Basis for \( P_3 \):** A basis of a vector space is a set of linearly independent vectors that span the full space. Candidates for the basis of \( P_3 \) should include a combination of terms that cover all necessary degrees (i.e., from 0 to 3).
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**Note:** Learners should attempt to identify the correct basis by checking each set for linear independence and ensuring that they span all polynomials in \( P_3 \).](https://content.bartleby.com/qna-images/question/5025798a-663c-4f82-a82a-d0677c58328a/fa694982-3ead-4af9-b5ac-245ca97897a1/w144pjne_processed.png)
Transcribed Image Text:**Transcription for Educational Website**
---
**Problem: Understanding the Vector Space of Polynomials**
1. **Find the Dimension of \( P_3 \):**
What dimension is \( P_3 \) (the set of polynomials of degree 3 or less)?
[Input Box for Answer]
2. **Determine a Basis for \( P_3 \):**
Which of the following sets form a basis for \( P_3 \)?
- \(\{2x^2, x, 0, x^3\}\)
- \(\{2 + x^3, -2x^2, x, -5, 2x + 3\}\)
- \(\{3, -x^3, x\}\)
- \(\{1, 2x, -3x^2, 2x^3\}\)
- \(\{2x^2, x, -5, 2 + x^3\}\)
**Explanation:**
- **Dimension of \( P_3 \):** In linear algebra, the dimension of a vector space is the number of vectors in a basis for the space. For the space \( P_3 \), the basis is typically \(\{1, x, x^2, x^3\}\), indicating a dimension of 4.
- **Basis for \( P_3 \):** A basis of a vector space is a set of linearly independent vectors that span the full space. Candidates for the basis of \( P_3 \) should include a combination of terms that cover all necessary degrees (i.e., from 0 to 3).
---
**Note:** Learners should attempt to identify the correct basis by checking each set for linear independence and ensuring that they span all polynomials in \( P_3 \).
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