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T(S)-(S(1)." (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the evaluation of d'f the third derivative of fat 1, ie.. d (a) Show that T is a linear map. (c) Calculate the matrix representing 7 with respect to the standard basis e, (1.0)". e(0, 1) of R² and the basis (v₁. V2. Va. Va. Vs) of P(R). (b) Show that the set (V₁, V2, V3, V4. Vs)= (1+2,1-₁² + x³, 1-x²₁x²+x¹) is a basis for P(R). (d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁ Vs and state the dimension of the kernel, Ker(7), of T.

T(S)-(S(1)." (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the evaluation of d'f the third derivative of fat 1, ie.. d (a) Show that T is a linear map. (c) Calculate the matrix representing 7 with respect to the standard basis e, (1.0)". e(0, 1) of R² and the basis (v₁. V2. Va. Va. Vs) of P(R). (b) Show that the set (V₁, V2, V3, V4. Vs)= (1+2,1-₁² + x³, 1-x²₁x²+x¹) is a basis for P(R). (d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁ Vs and state the dimension of the kernel, Ker(7), of T.

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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T(S)-(S(1),f" (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the evaluation of the third derivative of f at 1, i.e. (a) Show that T is a linear map. (c) Calculate the matrix representing 7 with respect to the standard basis e,= (1,0). e(0, 1) of R² and the basis (v₁, V2, Vs. Va. Vs) of P, (R). (b) Show that the set (V₁, V2, V3, V₁, Vs) = (1+z,1-2,²+³,1-¹,²+¹) is a basis for P(R). (d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁..., Vs and state the dimension of the kernel, Ker(7), of T.
Transcribed Image Text:T(S)-(S(1),f" (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the evaluation of the third derivative of f at 1, i.e. (a) Show that T is a linear map. (c) Calculate the matrix representing 7 with respect to the standard basis e,= (1,0). e(0, 1) of R² and the basis (v₁, V2, Vs. Va. Vs) of P, (R). (b) Show that the set (V₁, V2, V3, V₁, Vs) = (1+z,1-2,²+³,1-¹,²+¹) is a basis for P(R). (d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁..., Vs and state the dimension of the kernel, Ker(7), of T.
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