Answered: Find a basis for the following subsapce…

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 the dimmensions of the following vector spaces
Let P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = { 1, (x-1), (x-1)2, (x-1)3}. Show that B is a basis for P3[x]. Give p(x) = x3+ x, find [p(x)]B.
Find a basis for the vector space of all quadratic polynomials p(x) = ax²+bx+c, that satisfy p(−1) = 0.
Let P₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 14x² − 17x + 10, 17x – 19x² – 11 and 9x² - 10x + 6. a. The dimension of the subspace H is b. Is {14x² - 17x + 10, 17x - 19x² – 11,9x² – 10x +6} a basis for P₂? choose c. A basis for the subspace H is { (where you can enter xx in place of x²) Be sure you can explain and justify your answer. }. Enter a polynomial or a comma separated list of polynomials
Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by - 3x + 2, 12x – 7x2 – 2 and 3x2 – 3x + 1. 5x2 a. The dimension of the subspace H is b. Is {5æ? . - 3x + 2, 12x – 7x? – 2, 3x? – 3x + 1} a basis for P2? c. A basis for the subspace H is {
Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of p(x)=2+1.7x equals: Answer:
Let ₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 7x - 2x² + 2, ·(x² + 2x +2) and 3x − x² + 1. 1 a. The dimension of the subspace H is b. Is {7x − 2x² +2,- (x² + 2x +2), 3x − x² +1} a basis for ➡? choose Be sure you can explain and justify your answer. c. A basis for the subspace His { Enter a polynomial or a comma separated list of polynomials (where you can enter xx in place of x²)
Let P₂ be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial p(x) = x+x²-2 relative to the basis B = {1, x -2, x²-x}.
Consider the space P₂ which is the vector space of polynomials of degree not greater than two. We define the inner product in P2 according to the formula -Lpqdz. Also, consider the following two vectors (a) Calculate the inner product (p, q) (b) Calculate the distance between vectors p and q. (c) Calculate the norm ||q||, of vector q. (p, q) (e) For vectors p and q, calculate the inner product (p - q, p + 2q). p = x² = 5x + 1, q= −2x²+x. (d) For vectors p and q, find the orthogonal projection of p on q (projp). Your answer must be a vector from P2, i.e. a polynomial of degree 2 or less.

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Find a basis for the following subsapce of the vector space of polynomials of degree ≤ 3: W: = span {1+x+x2+x3, -1+5x+x2+11zx3, -1+2x+5x3, 1+6x+4x2+x3}