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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed029406-a1c1-473f-a3a0-6fd0fbd8e89d%2F9ca4905c-dfb7-44a8-af19-3a77eb561d29%2Fd6fpnpe_processed.jpeg&w=3840&q=75)
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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