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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 4EQ
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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? ?
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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