Define T:P,→R3 as shown to the right.P(-1)Find the image under T of p(t) = -2-tT(p) =p(0)b. Show that Tis a linear transformation.p(1)c. Find the matrix for T relative to the basis B= (b,, b,, ba) = (1, t, t2} for P, and the standard basis E= (e,, e,, eg) for R3.-----a. The image under Tof p(t) = -2-t is.b. Let p(t) and q(t) be polynomials in P2. Show that T(p(t) + q(t)) = T(p(t)) + T(q(t)). First apply the definition of T.(p+ g(1)T(p(t) + q(t)) =(р+ g)(0)Next apply the definition of (p + q)(t). What is the result?(p + q)(- 1)P(- 1)+ q(1)p(t) + q(t)(p+ qXt)OA.p(0) + q(0)OB.p(t) + q(t)(p + g)(t)P(1) + q(- 1)p(t) + q(t)(p+ q)(t)tis tip(1) + q(1)p(- 1) + q(- 1)(p + q)( - 1)Oc.p(0) + q(0)OD.p(0) + q(0)(p + q)(0)P(- 1)+ q(- 1)p(1) + q(1)(p+ q)(1)Rewrite this as the sum of two vectors. What is the result?p(t)q(t)P(- 1)q(- 1)O A.p(t)q(t)OB.P(0)q(0)p(t)q(t)P(1)q(1)P(1)q(1)P(-1)q(1)OC.p(0)q(0)OD.p(0)q(0)p(- 1)q( - 1)P(1)q(- 1) Now apply the definition of T again. What is the result?O A. p(t) + T(q(t))O B. T(p(t)) + T(q(t))OC. T(p(t)) + q(t)Let p(t) be a polynomial in P, and let c be a scalar. Show that T(c• (t)) = c• T(p(t)). First apply the definition of T.T(c• p() =Next apply the definition of (c. p)(t). What is the result?c•p(1)c• p(t)c•P(-1)OA.c• p(0)OB.c•p(0)Oc.C•p(0)C*P(-1)c•p(t)c•p(1)Remove a common factor from this vector. What is the result?p(t)p(1)P(- 1)O A. C p(t)о в. с.p(0)OC. C*p(0)P(t)p(-1)P(1)Now apply the definition of T again, thus completing the proof that Tis a linear transformation. what is the result?O A. T(p(t)) +cO B. c.T(p(t))О С. Т(р())c. The matrix for T relative to B and E is

Answered: Image /qna-images/answer/48a840a2-1833-4b92-8714-28f744fda710.jpg

Define T:P,-R as shown to the right. P(- 1) a. Find the image under T of p(t) = - 2-t b. Show that Tis a linear transformation. c. Find the matrix for T relative to the basis B= (b,, b2. b) = (1, t, ) for Pz and the standard basis E= (e,. e2. e3) for R. T(p) = P(0) P(1) a. The image under Tof p() -2-tis b. Let p(t) and q(t) be polynomials in Pz. Show that T(p() + q() = T(p() + T(q(). First apply the definition of T. (p•9)(1) T(p() + q() = (p+ aX0) Next apply the definition of (p + glt), What is the result? (p• q)(- 1) P(- 1)+ q(1) P(t) • q(t) (p+aX1) O A. P(0) + q(0) OB. P() + q() (p+ q)() p(1)+ al- 1) P(1) + q(t) tis ti P(1) • q(1) P(- 1)• q(- 1) (p+aX- 1) Oc. P(0) + q(0) OD. P(0) + q(0) (p+ gX0) p(- 1)+ q(- 1) P(1) + q(1) (p+ aX1) Rewrite this as the sum of two vectors. What is the result? P(1) P(- 1) 9(- 1) O A. P() OB. P(0) q(0) P(t) P(1) 9(1) P(1) q(1) P(- 1) 9(1) Oc. P(0) q(0) OD. P(0) q(0) P(- 1) 9- 1) P(1) q( - 1)

Define T:P,-R as shown to the right. P(- 1) a. Find the image under T of p(t) = - 2-t b. Show that Tis a linear transformation. c. Find the matrix for T relative to the basis B= (b,, b2. b) = (1, t, ) for Pz and the standard basis E= (e,. e2. e3) for R. T(p) = P(0) P(1) a. The image under Tof p() -2-tis b. Let p(t) and q(t) be polynomials in Pz. Show that T(p() + q() = T(p() + T(q(). First apply the definition of T. (p•9)(1) T(p() + q() = (p+ aX0) Next apply the definition of (p + glt), What is the result? (p• q)(- 1) P(- 1)+ q(1) P(t) • q(t) (p+aX1) O A. P(0) + q(0) OB. P() + q() (p+ q)() p(1)+ al- 1) P(1) + q(t) tis ti P(1) • q(1) P(- 1)• q(- 1) (p+aX- 1) Oc. P(0) + q(0) OD. P(0) + q(0) (p+ gX0) p(- 1)+ q(- 1) P(1) + q(1) (p+ aX1) Rewrite this as the sum of two vectors. What is the result? P(1) P(- 1) 9(- 1) O A. P() OB. P(0) q(0) P(t) P(1) 9(1) P(1) q(1) P(- 1) 9(1) Oc. P(0) q(0) OD. P(0) q(0) P(- 1) 9- 1) P(1) q( - 1)

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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Question

Define T:P,→R3 as shown to the right.
P(-1)
Find the image under T of p(t) = -2-t
T(p) =
p(0)
b. Show that Tis a linear transformation.
p(1)
c. Find the matrix for T relative to the basis B= (b,, b,, ba) = (1, t, t2} for P, and the standard basis E= (e,, e,, eg) for R3.
-----
a. The image under Tof p(t) = -2-t is.
b. Let p(t) and q(t) be polynomials in P2. Show that T(p(t) + q(t)) = T(p(t)) + T(q(t)). First apply the definition of T.
(p+ g(1)
T(p(t) + q(t)) =
(р+ g)(0)
Next apply the definition of (p + q)(t). What is the result?
(p + q)(- 1)
P(- 1)+ q(1)
p(t) + q(t)
(p+ qXt)
OA.
p(0) + q(0)
OB.
p(t) + q(t)
(p + g)(t)
P(1) + q(- 1)
p(t) + q(t)
(p+ q)(t)
tis ti
p(1) + q(1)
p(- 1) + q(- 1)
(p + q)( - 1)
Oc.
p(0) + q(0)
OD.
p(0) + q(0)
(p + q)(0)
P(- 1)+ q(- 1)
p(1) + q(1)
(p+ q)(1)
Rewrite this as the sum of two vectors. What is the result?
p(t)
q(t)
P(- 1)
q(- 1)
O A.
p(t)
q(t)
OB.
P(0)
q(0)
p(t)
q(t)
P(1)
q(1)
P(1)
q(1)
P(-1)
q(1)
OC.
p(0)
q(0)
OD.
p(0)
q(0)
p(- 1)
q( - 1)
P(1)
q(- 1)

Now apply the definition of T again. What is the result?
O A. p(t) + T(q(t))
O B. T(p(t)) + T(q(t))
OC. T(p(t)) + q(t)
Let p(t) be a polynomial in P, and let c be a scalar. Show that T(c• (t)) = c• T(p(t)). First apply the definition of T.
T(c• p() =
Next apply the definition of (c. p)(t). What is the result?
c•p(1)
c• p(t)
c•P(-1)
OA.
c• p(0)
OB.
c•p(0)
Oc.
C•p(0)
C*P(-1)
c•p(t)
c•p(1)
Remove a common factor from this vector. What is the result?
p(t)
p(1)
P(- 1)
O A. C p(t)
о в. с.
p(0)
OC. C*
p(0)
P(t)
p(-1)
P(1)
Now apply the definition of T again, thus completing the proof that Tis a linear transformation. what is the result?
O A. T(p(t)) +c
O B. c.T(p(t))
О С. Т(р())
c. The matrix for T relative to B and E is

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