Which of the following is (are) a basis of P (R), the space of polynomials with real coefficients of d at most n? (Choose all that apply) O{n} O {1, x,x²,...,x}(degree of increases by 1 up to m) ○ {1, x,x²,..., O {x, x²,..., ,a"}(degree of x increases by 1 up to n) O {2, x, 3x²,..., ,2-1} (degree of increases by 1 up to n - 1) O [25, x, x¹,...,34"}(degree of increases up to 1) O {5, 2x, x², ..., 9a} (degree of increases by 1 up to n) -¹} (degree of increases by 1 up to n − 1)
Which of the following is (are) a basis of P (R), the space of polynomials with real coefficients of d at most n? (Choose all that apply) O{n} O {1, x,x²,...,x}(degree of increases by 1 up to m) ○ {1, x,x²,..., O {x, x²,..., ,a"}(degree of x increases by 1 up to n) O {2, x, 3x²,..., ,2-1} (degree of increases by 1 up to n - 1) O [25, x, x¹,...,34"}(degree of increases up to 1) O {5, 2x, x², ..., 9a} (degree of increases by 1 up to n) -¹} (degree of increases by 1 up to n − 1)
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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![**Question:**
Which of the following is (are) a basis of \( P_n(\mathbb{R}) \), the space of polynomials with real coefficients of degree at most \( n \)? (Choose all that apply)
- [ ] \(\{x^n\}\)
- [ ] \(\{1, x, x^2, \ldots, x^n\}\) (degree of \( x \) increases by 1 up to \( n \))
- [ ] \(\{1, x, x^2, \ldots, x^{n-1}\}\) (degree of \( x \) increases by 1 up to \( n-1 \))
- [ ] \(\{x, x^2, \ldots, x^n\}\) (degree of \( x \) increases by 1 up to \( n \))
- [ ] \(\{2, x, 3x^2, \ldots, 2x^{n-1}\}\) (degree of \( x \) increases by 1 up to \( n-1 \))
- [ ] \(\{2x, x^2, x^4, \ldots, 34x^n\}\) (degree of \( x \) increases up to \( n \))
- [ ] \(\{5, 2x, x^2, \ldots, 9x^n\}\) (degree of \( x \) increases by 1 up to \( n \))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F50a43fb0-bf84-47f9-92fb-e4e959a9d4e9%2Fcwyoy0q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Which of the following is (are) a basis of \( P_n(\mathbb{R}) \), the space of polynomials with real coefficients of degree at most \( n \)? (Choose all that apply)
- [ ] \(\{x^n\}\)
- [ ] \(\{1, x, x^2, \ldots, x^n\}\) (degree of \( x \) increases by 1 up to \( n \))
- [ ] \(\{1, x, x^2, \ldots, x^{n-1}\}\) (degree of \( x \) increases by 1 up to \( n-1 \))
- [ ] \(\{x, x^2, \ldots, x^n\}\) (degree of \( x \) increases by 1 up to \( n \))
- [ ] \(\{2, x, 3x^2, \ldots, 2x^{n-1}\}\) (degree of \( x \) increases by 1 up to \( n-1 \))
- [ ] \(\{2x, x^2, x^4, \ldots, 34x^n\}\) (degree of \( x \) increases up to \( n \))
- [ ] \(\{5, 2x, x^2, \ldots, 9x^n\}\) (degree of \( x \) increases by 1 up to \( n \))
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