can someone answer this STEP BY STEP and show all their work with calculation and everything please:)) (Differential Equations with Linear Algebra) Consider the following linear transformation:T: P2(R) → R³T(p(x)) = (p(0), p(2),p'(1))(a) Evaluate T(x² + 3x + 1)(b) Determine the Ker(T), a basis for the Ker(T), and the dimension of Ker(T).(c) Determine the Rng(T), a basis for the Rng(T), and the dimension of Rng(T).(d) Is T a 1-to-1 transformation? Is T an onto transformation?

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Answered: Consider the following linear…

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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can someone answer this STEP BY STEP and show all their work with calculation and everything please:)) (Differential Equations with Linear Algebra)

Consider the following linear transformation:
T: P2(R) → R³
T(p(x)) = (p(0), p(2),p'(1))
(a) Evaluate T(x² + 3x + 1)
(b) Determine the Ker(T), a basis for the Ker(T), and the dimension of Ker(T).
(c) Determine the Rng(T), a basis for the Rng(T), and the dimension of Rng(T).
(d) Is T a 1-to-1 transformation? Is T an onto transformation?

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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