:| Let T : P → R , such that T (ao +ax+a2x2 +a3r)-4 +ai +a, +a3 . a) Prove that T is a linear transformation b) Find the rank and nullity of T. c) Find a basis for the kernel of T.
:| Let T : P → R , such that T (ao +ax+a2x2 +a3r)-4 +ai +a, +a3 . a) Prove that T is a linear transformation b) Find the rank and nullity of T. c) Find a...
a. find the form of Kernel(T)
b. find the form of range(T)
P2 be the mapping defined by 23. Let T: P -
P2 be the mapping defined by 23. Let T: P -
In the following transformations:
a)Find the Kernel and Image
b)Find dimK(T) and dimI(T) and show that
dimK(T)+dimi(T)=dimV
c)say if the transformations are injective, suprajective or
bijective
i)
such that:
ii),
such that:
iii)
, such that:
T:R? → R Tx,y,z) = (x - y, 22) T: M2:3 (R) → M2.2(R) SZD LED TEDT SID ZID IIDL sip-sip tip-tip) = T: P2(R) + P(R) T(p2)) = rp() +p (2)
2) Let Let T : R3 - R3 such that T(ij) ,, j 1,2,3. Find the matrix A associated to T in the canonical basis. Find a basis of its kernel and its image. Verify your answers.
2) Let Let T : R3 - R3 such that T(ij) ,, j 1,2,3. Find the matrix A associated to T in the canonical basis. Find a basis of its kernel and its image. Verify your answers.
Let T:P R^2 be defined by T(p(x)) = (p(1),p(-1)). (a) Find T(p(x)) where p(x) = 2 + 5x. (b) Show that T is a linear transformation. (C) Find the kernel of T. Explain why T is one-to-one. (d) Find the range of T. Explain why I' is onto. (e) Find T-1(3,7)
Let f:04 →U14 be defined by f(x) = x². Find the kernel off and show that it is a normal subgroup of U14. Show detail step of solution.
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for the nullspace (Kernel) of T. c) Find a basis for the range of T.
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for...
n) Let T be a linear transformation given b : Find the standard matrix of T, and uce to find bases for the range of T (RCT) and the kernel of CkerCT)). What is tle domain of T? Codomein? coutd
n) Let T be a linear transformation given b : Find the standard matrix of T, and uce to find bases for the range of T (RCT) and the kernel of CkerCT)). What is tle domain of T? Codomein? coutd
10. Let T : P P , be the linear transformation defined by T(P) = (a) What is the kernel of T? (b) According to the concept of the rank theorem, what is the dimension of the range of T? (C) (needs an idea from earlier in the semester) If we represent P, by coordinate vectors rela- tive to it's standard basis (1.1.1-.1') and P, by coordinate vectors relative to it's standard basis (1,1,1"), find the standard matrix A of...
Answer to (a) is image = Z2 • {0,2} (where • is the external
direct product). And the kernel is {e,r^2} (where r is the
rotation). Answer to (c) is isomorphic to Z2 • Z2. Please show
work. I’m given answers but need to see how to get there.
Thanks
(20 poiants) Amer aocat (a) (5 points) Identify the kernel and image of the homomorphism from D, to Z2 Z1 (the infinite cyclic group) given by the rules p(r) (1,0...