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24. Let W be the subset of P3 defined by W = {p(x) in P3: p(1) = 0 and p'(-1)=0}. Show that W is a subspace of P3, and find a spanning set for W. 25. Find a spanning set for each of the subsets that is a subspace in Exercises 1-8. 26. Show that the set W of all symmetric (3 × 3) matri- ces is a subspace of the vector space of all (3 x 3) matrices. Find a spanning set for W. = 27. The trace of an (n x n) matrix A (aij), denoted tr(A), is defined to be the sum of the diagonal el- ements of A; that is, tr(A) = a₁ + a22+...+ ann. Let V be the vector space of all (3 x 3) matrices, and let W be defined by W = {A in V: tr(A) = 0}. Show that W is a subspace of V, and exhibit a span- ning set for W. 28. Let A be an (n × n) matrix. Show that B = (A+AT)/2 is symmetric and that C = (A - A¹)/2 is skew symmetric. 29. Use Exercise 28 to show that every (n x n) matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 30. Use Exercises 26 and 29 and Example 7 to construct a spanning set for the vector space of all (3 x 3) matrices where the spanning set consists entirely of

24. Let W be the subset of P3 defined by W = {p(x) in P3: p(1) = 0 and p'(-1)=0}. Show that W is a subspace of P3, and find a spanning set for W. 25. Find a spanning set for each of the subsets that is a subspace in Exercises 1-8. 26. Show that the set W of all symmetric (3 × 3) matri- ces is a subspace of the vector space of all (3 x 3) matrices. Find a spanning set for W. = 27. The trace of an (n x n) matrix A (aij), denoted tr(A), is defined to be the sum of the diagonal el- ements of A; that is, tr(A) = a₁ + a22+...+ ann. Let V be the vector space of all (3 x 3) matrices, and let W be defined by W = {A in V: tr(A) = 0}. Show that W is a subspace of V, and exhibit a span- ning set for W. 28. Let A be an (n × n) matrix. Show that B = (A+AT)/2 is symmetric and that C = (A - A¹)/2 is skew symmetric. 29. Use Exercise 28 to show that every (n x n) matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 30. Use Exercises 26 and 29 and Example 7 to construct a spanning set for the vector space of all (3 x 3) matrices where the spanning set consists entirely of

Advanced Engineering Mathematics
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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Linear algebra: please solve q24 and 31 correctly and handwritten 

24. Let W be the subset of P3 defined by W = {p(x) in P3: p(1) = 0 and p'(-1) = 0}. Show that W is a subspace of P3, and find a spanning set for W. 25. Find a spanning set for each of the subsets that is a subspace in Exercises 1-8. 26. Show that the set W of all symmetric (3 × 3) matri- ces is a subspace of the vector space of all (3 × 3) matrices. Find a spanning set for W. = 27. The trace of an (n x n) matrix A (aij), denoted tr(A), is defined to be the sum of the diagonal el- ements of A; that is, tr(A) = a₁ + a22+...+ ann. Let V be the vector space of all (3 × 3) matrices, and let W be defined by W = {A in V: tr(A) = 0}. Show that W is a subspace of V, and exhibit a span- ning set for W. 28. Let A be an (n × n) matrix. Show that B = (A+AT)/2 is symmetric and that C = (A - A¹)/2 is skew symmetric. 29. Use Exercise 28 to show that every (n = n) matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 30. Use Exercises 26 and 29 and Example 7 to construct a spanning set for the vector space of all (3 x 3) matrices where the spanning set consists entirely of symmetric and skew-symmetric matrices. Specify how a (3 x 3) matrix A = (aij) can be expressed by using this spanning set. 31. Let V be the set of all (3 × 3) upper-triangular ma- trices, and note that V is a vector space. Each of the subsets W is a subspace of V. Find a spanning set for W. a) W = {A in V: a₁1 = 0, a22 = 0, a33 = 0} b) W = {A in V: a₁1 + a22+a33 = 0, a12 + a23 = 0}
Transcribed Image Text:24. Let W be the subset of P3 defined by W = {p(x) in P3: p(1) = 0 and p'(-1) = 0}. Show that W is a subspace of P3, and find a spanning set for W. 25. Find a spanning set for each of the subsets that is a subspace in Exercises 1-8. 26. Show that the set W of all symmetric (3 × 3) matri- ces is a subspace of the vector space of all (3 × 3) matrices. Find a spanning set for W. = 27. The trace of an (n x n) matrix A (aij), denoted tr(A), is defined to be the sum of the diagonal el- ements of A; that is, tr(A) = a₁ + a22+...+ ann. Let V be the vector space of all (3 × 3) matrices, and let W be defined by W = {A in V: tr(A) = 0}. Show that W is a subspace of V, and exhibit a span- ning set for W. 28. Let A be an (n × n) matrix. Show that B = (A+AT)/2 is symmetric and that C = (A - A¹)/2 is skew symmetric. 29. Use Exercise 28 to show that every (n = n) matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 30. Use Exercises 26 and 29 and Example 7 to construct a spanning set for the vector space of all (3 x 3) matrices where the spanning set consists entirely of symmetric and skew-symmetric matrices. Specify how a (3 x 3) matrix A = (aij) can be expressed by using this spanning set. 31. Let V be the set of all (3 × 3) upper-triangular ma- trices, and note that V is a vector space. Each of the subsets W is a subspace of V. Find a spanning set for W. a) W = {A in V: a₁1 = 0, a22 = 0, a33 = 0} b) W = {A in V: a₁1 + a22+a33 = 0, a12 + a23 = 0}
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