Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to 2. Let B = {1, x, x2} be an ordered basis for P2. Let [ ] B : P2 linear transformation determined by [1] B =1, [x]B=2, [x2]B=e3. → R³ be the Find the coordinate vector representation for each of the following polynomials. Your answers should be vectors of the general form <1,2,3>. a. [-5]g b. [-6+x²] B = c. [x2-3x+2] B == d. Is the linear transformation ( ) B an isomorphism? choose
Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to 2. Let B = {1, x, x2} be an ordered basis for P2. Let [ ] B : P2 linear transformation determined by [1] B =1, [x]B=2, [x2]B=e3. → R³ be the Find the coordinate vector representation for each of the following polynomials. Your answers should be vectors of the general form <1,2,3>. a. [-5]g b. [-6+x²] B = c. [x2-3x+2] B == d. Is the linear transformation ( ) B an isomorphism? choose
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 P2 denote the vector space of all polynomials in the variable x of degree less than
or equal to 2. Let B = {1, x, x2} be an ordered basis for P2. Let [ ] B : P2
linear transformation determined by
[1] B = 1, [x]B=2, [x2]B=e3.
→ R³ be the
Find the coordinate vector representation for each of the following polynomials.
Your answers should be vectors of the general form <1,2,3>.
a. [-5]g
b. [-6+x²] B
=
c. [x2 3x+2] B
=
d. Is the linear transformation ( ) B an isomorphism? choose](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5f3a7f3-10c4-4869-906a-40675d8d17a6%2Fc48691d4-4510-4d5f-ab0b-32be69afe626%2Ftel7t7f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let P2 denote the vector space of all polynomials in the variable x of degree less than
or equal to 2. Let B = {1, x, x2} be an ordered basis for P2. Let [ ] B : P2
linear transformation determined by
[1] B = 1, [x]B=2, [x2]B=e3.
→ R³ be the
Find the coordinate vector representation for each of the following polynomials.
Your answers should be vectors of the general form <1,2,3>.
a. [-5]g
b. [-6+x²] B
=
c. [x2 3x+2] B
=
d. Is the linear transformation ( ) B an isomorphism? choose
Transcribed Image Text:Let P3 be the vector space of all polynomials of degree 3 or less in the variable x. Let
P1(x)
= -1+x-x²,
P2(x) =
-3+3x-3x²,
P3(x)
=
-
1 2x + x2,
P4(x)
=
1-4x+2x2
and let C {p₁(x), P2(x), P3(x), P4(x)}.
=
a. Use coordinate representations with respect to the basis B = {1, x, x², x³} to
determine whether the set C forms a basis for P3. choose
b. Find a basis for span(C). Enter a polynomial or a comma separated list of
polynomials.
{
}
c. The dimension of span(C) is
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