Consider the map T: P2 → P1 given by T(ao + aj x + az x?) = (3a0 – 3a,) – 2a7 x. %3D (a) Calculate T(2x – x²) = (b) Give an example of a non-zero polynomial p(x) E P2 so that T (p(x)) = 0 p(x) = (c) Does T preserve vector addition? ? If not, enter two polynomials p(x), q(x) E P2 such that T(p + q) # T(p) +T(q). Otherwise leave the following spaces blank.
Consider the map T: P2 → P1 given by T(ao + aj x + az x?) = (3a0 – 3a,) – 2a7 x. %3D (a) Calculate T(2x – x²) = (b) Give an example of a non-zero polynomial p(x) E P2 so that T (p(x)) = 0 p(x) = (c) Does T preserve vector addition? ? If not, enter two polynomials p(x), q(x) E P2 such that T(p + q) # T(p) +T(q). Otherwise leave the following spaces blank.
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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Transcribed Image Text:Consider the map T: P2 → Pi given by
T(ao + aj x + azx²) = (3a0 – 3az)² – 2a; x.
(a) Calculate T(2x – x²) =
(b) Give an example of a non-zero polynomial p(x) E P2 so that T (p(x)) = 0
p(x) =
(c) Does T preserve vector addition? ?
If not, enter two polynomials p(x), q(x) E P2 such that
T(p + q) + T(p) + T(q).
Otherwise leave the following spaces blank.
p(x) =
q(x) =
(d) Does T preserve scalar multiplication? ?
If not, enter a scalar k and polynomial r e P2 such that
T(kr) + kT(r).
Otherwise leave the following spaces blank.
k =
and r(x) =
(e) Is T a linear transformation? ?
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