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Let p, (1) = 2 +?, P2(t) =t-4², p3(t) = 3 +t- 3r?. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P2. What are the coordinate vectors corresponding to p1, P2, and p3? P1 = . P2 = P3 Place these coordinate vectors into the columns of a matrix A. What can be said about the matrix A? O A. The matrix A is invertible because it is row equivalent to lą and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to lą and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A is invertible because it is row equivalent to lz and therefore the original columns of A form a basis for R° by the Invertible Matrix Theorem. D. The matrix A forms a basis for R° by the Invertible Matrix Theorem because all square matrices are row equivalent to l3. How does this show that the polynomials form a basis for P2? O A. The polynomials form a basis for P, because of the isomorphism between R and P,. B. The polynomials form a basis for P2 because the matrix A is invertible. OC. The polynomials form a basis for P2 because the columns of A are linearly independent. D. The polynomials form a basis for P, because any basis in R" is also a basis in P,. -2 b. Consider the basis B= {p1, P2, P3} for P2. Find q in P2, given that [q]g = 1 2 q(1) = (O + Ot+ O?

Let p, (1) = 2 +?, P2(t) =t-4², p3(t) = 3 +t- 3r?. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P2. What are the coordinate vectors corresponding to p1, P2, and p3? P1 = . P2 = P3 Place these coordinate vectors into the columns of a matrix A. What can be said about the matrix A? O A. The matrix A is invertible because it is row equivalent to lą and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to lą and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A is invertible because it is row equivalent to lz and therefore the original columns of A form a basis for R° by the Invertible Matrix Theorem. D. The matrix A forms a basis for R° by the Invertible Matrix Theorem because all square matrices are row equivalent to l3. How does this show that the polynomials form a basis for P2? O A. The polynomials form a basis for P, because of the isomorphism between R and P,. B. The polynomials form a basis for P2 because the matrix A is invertible. OC. The polynomials form a basis for P2 because the columns of A are linearly independent. D. The polynomials form a basis for P, because any basis in R" is also a basis in P,. -2 b. Consider the basis B= {p1, P2, P3} for P2. Find q in P2, given that [q]g = 1 2 q(1) = (O + Ot+ O?

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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Let p, (1) = 2 +?, P2(t) =t-4², p3(t) = 3 +t- 3r?. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P2. What are the coordinate vectors corresponding to p1, P2, and p3? P1 = . P2 = P3 Place these coordinate vectors into the columns of a matrix A. What can be said about the matrix A? O A. The matrix A is invertible because it is row equivalent to lą and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to lą and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A is invertible because it is row equivalent to lz and therefore the original columns of A form a basis for R° by the Invertible Matrix Theorem. D. The matrix A forms a basis for R° by the Invertible Matrix Theorem because all square matrices are row equivalent to l3. How does this show that the polynomials form a basis for P2? O A. The polynomials form a basis for P, because of the isomorphism between R and P,. B. The polynomials form a basis for P2 because the matrix A is invertible. OC. The polynomials form a basis for P2 because the columns of A are linearly independent. D. The polynomials form a basis for P, because any basis in R" is also a basis in P,. -2 b. Consider the basis B= {p1, P2, P3} for P2. Find q in P2, given that [q]g = 1 2 q(1) = (O + Ot+ O?
Transcribed Image Text:Let p, (1) = 2 +?, P2(t) =t-4², p3(t) = 3 +t- 3r?. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P2. What are the coordinate vectors corresponding to p1, P2, and p3? P1 = . P2 = P3 Place these coordinate vectors into the columns of a matrix A. What can be said about the matrix A? O A. The matrix A is invertible because it is row equivalent to lą and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to lą and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A is invertible because it is row equivalent to lz and therefore the original columns of A form a basis for R° by the Invertible Matrix Theorem. D. The matrix A forms a basis for R° by the Invertible Matrix Theorem because all square matrices are row equivalent to l3. How does this show that the polynomials form a basis for P2? O A. The polynomials form a basis for P, because of the isomorphism between R and P,. B. The polynomials form a basis for P2 because the matrix A is invertible. OC. The polynomials form a basis for P2 because the columns of A are linearly independent. D. The polynomials form a basis for P, because any basis in R" is also a basis in P,. -2 b. Consider the basis B= {p1, P2, P3} for P2. Find q in P2, given that [q]g = 1 2 q(1) = (O + Ot+ O?
Expert Solution
Step 1

The polynomials p1t, p2t and p3t belongs to the vector space P2 because

P2 contains all the polynomials of degree less than or equal to 2 with their

coefficients in ℝ.

We have to create a matrix A with the help of the coordinate vectors of the

given polynomials.

steps

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