Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Equations and inequalities describe the relationship between two mathematical expressions.
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Number 22 linear algebra
V and if T is a set of more than p vectors in V, mensional vector then T is linearly dependent. 20. a. R2 is a two-dimensional subspace of R'. b. The number of variables in the equation Ax =0 equals the dimension of Nul A. = = %} c. A vector space is infinite-dimensional if it is spanned by an infinite set. d. If dim V =n and if S spans V, then S is a basis of V. s in R' whose e. The only three-dimensional subspace of R' is R' itself. 21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and -12t + 813. These polynomials arise naturally in the study of certain important differential equations in mathematical physics.? Show that the first four Hermite polynomials form a basis of P3. 2 spanned by the subspace 22. The first four Laguerre polynomials are 1, 1-1,2-4t +1, and 6 18t +9t2-t. Show that these polynomials form a basis of P3. 23. Let B be the basis of P, consisting of the Hermite polynomi- als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find the coordinate vector of p relative to B. 24. Let B be the basis of P, consisting of the first three Laguerre polynomials listed in Exercise 22, and let p(t) = 5+5t - 2t2. Find the coordinate vector of p relative For the matrices to B. 25. Let S be a subset of an n-dimensional vector space V, and suppose S contains fewer than n vectors. Explain why S cannot span V. 26. Let H be an n-dimensional subspace of an n-dimensional vector space V. Show that H =V. 27. Explain why the space P of all polynomials is an infinite- dimensional space. 2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor and David C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Other sets of polynomials are discussed there, too. 3.
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Transcribed Image Text:V and if T is a set of more than p vectors in V, mensional vector then T is linearly dependent. 20. a. R2 is a two-dimensional subspace of R'. b. The number of variables in the equation Ax =0 equals the dimension of Nul A. = = %} c. A vector space is infinite-dimensional if it is spanned by an infinite set. d. If dim V =n and if S spans V, then S is a basis of V. s in R' whose e. The only three-dimensional subspace of R' is R' itself. 21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and -12t + 813. These polynomials arise naturally in the study of certain important differential equations in mathematical physics.? Show that the first four Hermite polynomials form a basis of P3. 2 spanned by the subspace 22. The first four Laguerre polynomials are 1, 1-1,2-4t +1, and 6 18t +9t2-t. Show that these polynomials form a basis of P3. 23. Let B be the basis of P, consisting of the Hermite polynomi- als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find the coordinate vector of p relative to B. 24. Let B be the basis of P, consisting of the first three Laguerre polynomials listed in Exercise 22, and let p(t) = 5+5t - 2t2. Find the coordinate vector of p relative For the matrices to B. 25. Let S be a subset of an n-dimensional vector space V, and suppose S contains fewer than n vectors. Explain why S cannot span V. 26. Let H be an n-dimensional subspace of an n-dimensional vector space V. Show that H =V. 27. Explain why the space P of all polynomials is an infinite- dimensional space. 2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor and David C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Other sets of polynomials are discussed there, too. 3.
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