Explanation Given: A linear transformation is given by T ( f ( t ) ) = ∫ 0 2 f ( t ) d t The basis of P 2 is given by β = ( 1 , t , t 2 ) Concept Used: Consider a linear transformation T from V to V, and let B be the matrix of T with respect to a basis β = ( f 1 , f 2 , ... , f n ) of V. Then B = [ ... ... [ T ( f 1 ) ] β ... [ T ( f n ) ] β ... ... ] The columns of B are the β -coordinate vectors of the transforms of the basis elements β = ( f 1 , f 2 , ... , f n ) of V. Definition of kernel of a transformation: Kernel of a matrix A is a solution to the equation A x → = 0 Basis of a matrix A is a vector space in which every element is written as linear combination of elements of matrix A . Definition of Image of a transformation: Non redundant vectors of a matrix formed the bases of Image of it. Isomorphism: A linear transformation T : A → B will be Isomorphic if dim ( A ) = dim ( B ) and ker T = { 0 } . For a linear transformation T : A → B A linear transformation will be Isomorphic if it is invertible or the matrix transformation of it will be invertible. A matrix will be invertible if the determinant of the matrix is non-singular. That is a matrix [ a b c d ] will be invertible if a d − b c ≠ 0 Rank of a transformation: Rank of T = dim ( Im T ) Calculation: Consider the given linear transformation T : P 2 → P 2 given by T ( f ( t ) ) = ∫ 0 2 f ( t ) d t Consider the given basis β = ( 1 , t , t 2 ) for P 2 . Now substitute 1 in the given mapping and express the result as a linear combination of the given basis. T ( 1 ) = ∫ 0 2 1 d t T ( 1 ) = [ t ] 0 2 T ( 1 ) = 2 T ( 1 ) = ( 2 ) 1 + ( 0 ) t + ( 0 ) t 2 Therefore, [ T ( f 1 ) ] β = [ 2 0 0 ] Now substitute t in the given mapping and express the result as a linear combination of the given basis. T ( t ) = ∫ 0 2 t d t T ( t ) = 1 2 [ t 2 ] 0 2 T ( t ) = 2 T ( t ) = ( 2 ) 1 + ( 0 ) t + ( 0 ) t 2 Therefore, [ T ( f 2 ) ] β = [ 2 0 0 ] Now substitute t 2 in the given mapping and express the result as a linear combination of the given basis. T ( t ) = ∫ 0 2 t 2 d t T ( t ) = 1 3 [ t 3 ] 0 2 T ( t ) = 8 3 T ( t ) = ( 8 3 ) 1 + ( 0 ) t + ( 0 ) t 2 Therefore, [ T ( f 3 ) ] β = [ 8 3 0 0 ] The coefficients of basis β = ( 1 , t , t 2 ) in each case will make the column vector for the matrix representation. Therefore, the matrix representation of the given linear transformation will be B = [ 2 2 8 3 0 0 0 0 0 0 ] Now let us compute the determinant of the above matrix to check whether the matrix is invertible. | B | = | 2 2 8 3 0 0 0 0 0 0 | = 0 [ R 2 ∼ R 3 ] Therefore, the matrix is not invertible, so the matrix is Isomorphic

Linear Algebra with Applications (2-Download)

5th Edition

ISBN: 9780321796974

Author: Otto Bretscher

Publisher: PEARSON

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Chapter 4.3, Problem 29E

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The matrix representation of the given linear transformation from 1 P2 to P2 . Check the Isomorphism of T, if T is not isomorphic then find the basis of Im T and ker T. Also find the rank of T.

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Chapter 4.3, Problem 28E

Chapter 4.3, Problem 30E

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Chapter 4 Solutions

Linear Algebra with Applications (2-Download)

Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...

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The matrix representation of the given linear transformation from 1 P 2 to P 2 . Check the Isomorphism of T, if T is not isomorphic then find the basis of Im T and ker T. Also find the rank of T.

Solution Summary: The author analyzes the matrix representation of a linear transformation from 1 P_2 to 2. If T is not isomorphic, find the basis of Im T and ker T.

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The matrix representation of the given linear transformation from 1 P2 to P2 . Check the Isomorphism of T, if T is not isomorphic then find the basis of Im T and ker T. Also find the rank of T.

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Chapter 4 Solutions