Let p, ()=2+P, P21)=t-4°, P3(1) =4 +t-2. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P, What are the coordinate vectors corresponding to P,, P2, and p3? P, = 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 l, and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to l, and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A forms a basis for R3 by the Invertible Matrix Theorem because all square matrices are row equivalent to l,. O D. The matrix A is invertible because it is row equivalent to l, and therefore the original columns of A form a basis for R3 by the Invertible Matrix Theorem. How does this show that the polynomials form a basis for P,? O A. The polynomials form a basis for P, because any basis in Rº is also a basis in P. OB. The polynomials form a basis for P, because the columns of A are linearly independent. O C. The polynomials form a basis for P, because the matrix A is invertible. O D. The polynomials form a basis for P, because of the isomorphism between R3 and P, -3 b. Consider the basis B= (p,. Pa Pa) for P, Find q in P2. given that (q]g = 2. q(t) =
Let p, ()=2+P, P21)=t-4°, P3(1) =4 +t-2. Complete parts (a) and (b) below. a. Use coordinate vectors to show that these polynomials form a basis for P, What are the coordinate vectors corresponding to P,, P2, and p3? P, = 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 l, and therefore the row reduced columns of A form a basis for R3 by the Invertible Matrix Theorem. O B. The matrix A is invertible because it is row equivalent to l, and therefore the null space of A, denoted Nul A, forms a basis for R3 by the Invertible Matrix Theorem. O C. The matrix A forms a basis for R3 by the Invertible Matrix Theorem because all square matrices are row equivalent to l,. O D. The matrix A is invertible because it is row equivalent to l, and therefore the original columns of A form a basis for R3 by the Invertible Matrix Theorem. How does this show that the polynomials form a basis for P,? O A. The polynomials form a basis for P, because any basis in Rº is also a basis in P. OB. The polynomials form a basis for P, because the columns of A are linearly independent. O C. The polynomials form a basis for P, because the matrix A is invertible. O D. The polynomials form a basis for P, because of the isomorphism between R3 and P, -3 b. Consider the basis B= (p,. Pa Pa) for P, Find q in P2. given that (q]g = 2. q(t) =
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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Help needed in part A p1 and last part b Only thanks in advance
![Let p, (1) = 2 +t°, P21)=t-4, P3(1)= 4 +t-2. Complete parts (a) and (b) below.
a. Use coordinate vectors to show that these polynomials form a basis for P,.
What are the coordinate vectors corresponding to P, , P2, and p3?
P =
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 la and therefore the row reduced columns of A form a basis for R³ by the Invertible Matrix Theorem.
O B. The matrix A is invertible because it is row equivalent to l, and therefore the null space of A, denoted Nul A, forms a basis for R³ by the Invertible Matrix Theorem.
O C. The matrix A forms a basis for R3 by the Invertible Matrix Theorem because all square matrices are row equivalent tol2.
O D. The matrix A is invertible because it is row equivalent to la and therefore the original columns of A form a basis for R3 by the Invertible Matrix Theorem.
How does this show that the polynomials form a basis for P2?
O A. The polynomials form a basis for P, because any basis in Rº is also a basis in P.
O B. The polynomials form a basis for P, because the columns of A are linearly independent.
O C. The polynomials form a basis for P, because the matrix A is invertible.
O D. The polynomials form a basis for P, because of the isomorphism between R3 and P,
b. Consider the basis B= {p,, P2 pa) for P2. Find q in P2. given that [q]g =
q(t) = (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ebeefb-4552-4ee2-a18c-1232c59f100f%2F4fd992d7-c697-453f-9626-d606d94d07f3%2Fb9xbbxg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let p, (1) = 2 +t°, P21)=t-4, P3(1)= 4 +t-2. Complete parts (a) and (b) below.
a. Use coordinate vectors to show that these polynomials form a basis for P,.
What are the coordinate vectors corresponding to P, , P2, and p3?
P =
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 la and therefore the row reduced columns of A form a basis for R³ by the Invertible Matrix Theorem.
O B. The matrix A is invertible because it is row equivalent to l, and therefore the null space of A, denoted Nul A, forms a basis for R³ by the Invertible Matrix Theorem.
O C. The matrix A forms a basis for R3 by the Invertible Matrix Theorem because all square matrices are row equivalent tol2.
O D. The matrix A is invertible because it is row equivalent to la and therefore the original columns of A form a basis for R3 by the Invertible Matrix Theorem.
How does this show that the polynomials form a basis for P2?
O A. The polynomials form a basis for P, because any basis in Rº is also a basis in P.
O B. The polynomials form a basis for P, because the columns of A are linearly independent.
O C. The polynomials form a basis for P, because the matrix A is invertible.
O D. The polynomials form a basis for P, because of the isomorphism between R3 and P,
b. Consider the basis B= {p,, P2 pa) for P2. Find q in P2. given that [q]g =
q(t) = (
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