1. Let V = R[x] be the vector space (over R) of all polynomials in r with real coefficients. Define the map T :V →V by T(f(x)) = xf(x). (i) Determine whether or not T is linear (provide a proof or counterexample, as appropriate). (ii) Determine whether or not T is injective (provide a proof or counterexample, as appropriate). (iii) Determine whether or not T is surjective (provide a proof or counterexample, as appropriate). (iv) Determine whether or not T is an isomorphism (explaining your answer). (v) Find ker(T). (vi) Find Im(T). (vii) Show that T has no eigenvectors.
1. Let V = R[x] be the vector space (over R) of all polynomials in r with real coefficients. Define the map T :V →V by T(f(x)) = xf(x). (i) Determine whether or not T is linear (provide a proof or counterexample, as appropriate). (ii) Determine whether or not T is injective (provide a proof or counterexample, as appropriate). (iii) Determine whether or not T is surjective (provide a proof or counterexample, as appropriate). (iv) Determine whether or not T is an isomorphism (explaining your answer). (v) Find ker(T). (vi) Find Im(T). (vii) Show that T has no eigenvectors.
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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please send handwritten solution for Q 1 Part iv v vi vii
![1. Let V = R[x] be the vector space (over R) of all polynomials in r with real
coefficients. Define the map T:V V by
T(f(r)) = r f(x).
(i) Determine whether or not T is linear (provide a proof or counterexample, as
appropriate).
(ii) Determine whether or not T is injective (provide a proof or counterexample,
as appropriate).
(iii) Determine whether or not T is surjective (provide a proof or counterexample,
as appropriate).
(iv) Determine whether or not T is an isomorphism (explaining your answer).
(v) Find ker(T).
(vi) Find Im(T).
(vii) Show that T has no eigenvectors.
2. Let
1
1
A =
-2 -3 1
3
(i) Find the row rank of A.
(ii) By finding a square submatrix of A of suitable size, show that the determi-
nantal rank of A is at least 2.
(iii) Use your answer to part (i) and any result from the lecture notes to find the
column rank and determinantal rank of A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7995244-39eb-45ef-9c72-d3729c0dbdab%2Fbd51c1ac-897b-42d5-99e9-70b9c68529c5%2F4430c9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let V = R[x] be the vector space (over R) of all polynomials in r with real
coefficients. Define the map T:V V by
T(f(r)) = r f(x).
(i) Determine whether or not T is linear (provide a proof or counterexample, as
appropriate).
(ii) Determine whether or not T is injective (provide a proof or counterexample,
as appropriate).
(iii) Determine whether or not T is surjective (provide a proof or counterexample,
as appropriate).
(iv) Determine whether or not T is an isomorphism (explaining your answer).
(v) Find ker(T).
(vi) Find Im(T).
(vii) Show that T has no eigenvectors.
2. Let
1
1
A =
-2 -3 1
3
(i) Find the row rank of A.
(ii) By finding a square submatrix of A of suitable size, show that the determi-
nantal rank of A is at least 2.
(iii) Use your answer to part (i) and any result from the lecture notes to find the
column rank and determinantal rank of A.
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