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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

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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 3EQ
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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
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
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
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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