Let Pn= {ao+a1x + a2x² + ...+ anx" | ao, a1, · · · an E R} Be the amount of allpolynomial of degree highest n. Form the operator (function) T : P2 → P2, p → T (p) = p' + p, where p' is the usual derivative of the polynomial p. Also form the multplication operator âu : P2 → P3, âæ (p) = xp. (For example it applies that T (x) =1+xoch â(x) = x²). (a) Explain in detail whyP, forming a vector space. Also explain why Sn = (1, x, x², - · , x") form a base for Pn. (we call this the standard base Pn).
Let Pn= {ao+a1x + a2x² + ...+ anx" | ao, a1, · · · an E R} Be the amount of allpolynomial of degree highest n. Form the operator (function) T : P2 → P2, p → T (p) = p' + p, where p' is the usual derivative of the polynomial p. Also form the multplication operator âu : P2 → P3, âæ (p) = xp. (For example it applies that T (x) =1+xoch â(x) = x²). (a) Explain in detail whyP, forming a vector space. Also explain why Sn = (1, x, x², - · , x") form a base for Pn. (we call this the standard base Pn).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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please send handwritten solution for Q1 part a
Note och meaning and
![1.
Let Pn = {ao+ a1x + a2x² +· ·+ anx" | ao, a1,
degree highest n. Form the operator (function) T: P2→ P2, p → T (p) = p' + p, where p' is the
usual derivative of the polynomial p. Also form the multplication operator î : P2→ P3, &(p) = xp.
(For example it applies that T (x) =1+x och & (x) = x²).
an E R} Be the amount of allpolynomial of
(a) Explain in detail why P, forming a yector space. Also explain why Sn = (1, x, x², . .. ,x") form a
base for Pn. (we call this the standard base Pn).
(b) Justify why ât och T are linear transformations.
(c) Determine the standard matrices [â] och [T] for ât and Tin the base Sn (for the appropriate n). Also
determine [â o T] and [T o â±]. It applies that t o T=To£?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6163c245-b98d-49a2-ba13-9e9b478b9cef%2Fe70d2cbd-c354-431a-af85-ea736acd8d6b%2Fpdujzct_processed.png&w=3840&q=75)
Transcribed Image Text:1.
Let Pn = {ao+ a1x + a2x² +· ·+ anx" | ao, a1,
degree highest n. Form the operator (function) T: P2→ P2, p → T (p) = p' + p, where p' is the
usual derivative of the polynomial p. Also form the multplication operator î : P2→ P3, &(p) = xp.
(For example it applies that T (x) =1+x och & (x) = x²).
an E R} Be the amount of allpolynomial of
(a) Explain in detail why P, forming a yector space. Also explain why Sn = (1, x, x², . .. ,x") form a
base for Pn. (we call this the standard base Pn).
(b) Justify why ât och T are linear transformations.
(c) Determine the standard matrices [â] och [T] for ât and Tin the base Sn (for the appropriate n). Also
determine [â o T] and [T o â±]. It applies that t o T=To£?
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