(d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁,..., V5 and state the dimension of the kernel, Ker(T), of T.
(d) Hence, find a basis for the kernel of T expressed as linear combinations of the vectors V₁,..., V5 and state the dimension of the kernel, Ker(T), 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
answer the question d,please.
![2. Let T: P₁(R) → R2 be the map given by
T()=(f(1),f"(1))
where P₁ (R) is the set of polynomials of degree 4 or less and f"" (1) is the evaluation of
d³f\
the third derivative of f at 1, i.e.,
dx³
(a) Show that I is a linear map.
(b) Show that the set
(c) Calculate the matrix representing T with respect to the standard basis e₁ = = (1,0)¹,
e₂ = (0, 1)¹ of R² and the basis {V₁, V2, V3, V4, V5} of P₁ (R).
(d) Hence, find a basis for the kernel of T expressed as linear combinations of the
vectors V₁,..., v5 and state the dimension of the kernel, Ker(T), of T.
(e) Find a basis for the image of T, Im(T), and state its dimension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2Fdcb533c0-2535-49af-9df7-242a7c6b9960%2Fmcg2bhc_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let T: P₁(R) → R2 be the map given by
T()=(f(1),f"(1))
where P₁ (R) is the set of polynomials of degree 4 or less and f"" (1) is the evaluation of
d³f\
the third derivative of f at 1, i.e.,
dx³
(a) Show that I is a linear map.
(b) Show that the set
(c) Calculate the matrix representing T with respect to the standard basis e₁ = = (1,0)¹,
e₂ = (0, 1)¹ of R² and the basis {V₁, V2, V3, V4, V5} of P₁ (R).
(d) Hence, find a basis for the kernel of T expressed as linear combinations of the
vectors V₁,..., v5 and state the dimension of the kernel, Ker(T), of T.
(e) Find a basis for the image of T, Im(T), and state its dimension.
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