Algèbre linéaire 1

domain at which the function assumes the value 0 If f: X Rn is a function from X to Rn, then ker(f) = fx 2X : f(x) = 0g: Notice that ker(f) is a subset of X Also, if T(x) = Ax is a linear transformation from Rm to Rn, then ker(T) (also denoted ker(A)) is the set of solutions to the equation Ax = 0

7 Homomorphisms and the First Isomorphism Theorem

The kernel of f is the subgroup kerf = n : 5n 0 (mod 20)g= f0,4,8,12,16 /Z 36 This is simply the cyclic group C 5 3 The map sgn : Snf1, 1ggiven by sgn(s) = (1 if s

Part III Homomorphism and Factor Groups

2 Then ker(∆) = {f ∈ D(R) : df dx = 0}, which is the set of all constant functions C 3 The coset of a function g ∈ D(R) is ˆ f ∈ D(R) : df dx = g′ ˙ = {g +c : c ∈ R} = g +C Corollary 13 15 Let ϕ : G −→ G′ be a homomorphism of groups Then ϕ is injective if and only if ker(ϕ) = {e} (Therefore, from now on, to check

Group Homomorphisms - Christian Brothers University

Ker = SL(2,R) (4) Let R[x] denote the group of all polynomials with real coecients under addition Let : R[x] R[x] be deﬁned by (f) = f0 The group operation preservation is simply “the derivative of a sum is the sum of the derivatives ” (f +g) = (f +g)0 = f0 +g0 = (f)+(g) Ker is the set of all constant polynomials

F13YR1 ABSTRACT ALGEBRA Lecture Notes: Part 5 1 The image

Then Ker(f) is the set of integers congruent to 0 moodulo 3, ie Ker(f)=3Z Lemma 3 Let f: G H be a homomorphism Then Ker(f) is a subgroup of G Proof

Math 110: Worksheet 3

(ii) if T(T(v)) = 0 for some v 2V, then T(v) = 0 (i) )(ii) Assume Ker(T) \Im(T) = f0gand suppose that T(T(v)) = 0 for some v 2V Note then that T(v) is the image under T of some element in V so it belongs to Im(T) Moreover, as T sends T(v) to 0, we have T(v) 2Ker(T) Thus, T(v) 2Ker(T) \Im(T) = f0gso T(v) = 0 (ii) )(i) Conversely, assume

MATH 110: LINEAR ALGEBRA FALL 2007/08 PROBLEM SET 7 SOLUTIONS

Solving this, we get x= y= z= 0 and so (x;y;z) = (0;0;0) and so ker(T) = f(0;0;0)g Hence Tis invertible by a result in the lectures (b) Find a formula for T 1

10 2 The Kernel and Range - Old Dominion University

10 2 The Kernel and Range DEF (→p 441, 443) Let L : V →W be a linear transformation Then (a) the kernel of L is the subset of V comprised of all vectors whose image is the zero vector:

Solution Outlines for Chapter 10 - Earlham College

set of pairs such that a b= 0, or f(a;a)ja2Zg Finally, to nd ˚ 1(3) observe that (3;0) maps to 3 Thus ˚ 1(3) = (3;0) + Ker˚= f(a+ 3;a)ja2Zg # 36: Suppose that there is a homomorphism ˚from Z Z to a group Gsuch that ˚((3;2)) = aand ˚((2;1)) = b Determine ˚((4;4)) in terms of aand b Assume that the operation of Gis addition