1 / 20

# 3.II. Homomorphisms - PowerPoint PPT Presentation

3.II. Homomorphisms. 3.II.1. Definition 3.II.2. Range Space and Nullspace. 3.II.1. Definition. Definition 1.1 : Homomorphism A function between vector spaces h : V → W that preserves the algebraic structure is a homomorphism or linear map . I.e.,.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about '3.II. Homomorphisms' - louvain

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

3.II.1. Definition

3.II.2.Range Space and Nullspace

Definition 1.1: Homomorphism

A function between vector spaces h: V → W that preserves the algebraic structure is a homomorphismor linear map. I.e.,

Example 1.2: Projection map

is a homomorphism.

by

π: R3 → R2

Proof:

by

by

Example 1.4: Zero Homomorphism

h: V → W by v 0

Example 1.5:Linear Map

by

g: R3 → R

is linear & a homomorphism.

is not linear & hence not a homomorphism.

h: R3 → R

since

is linear & a homomorphism.

is not linear & hence not a homomorphism.

Lemma 1.6: A homomorphism sends a zero vector to a zero vector.

Lemma 1.7:

Each is a necessary and sufficient condition for f : V → W to be a homomorphism:

1.

and

2.

Example 1.8:

by

is a homomorphism.

g: R2 → R4

Theorem 1.9: A homomorphism is determined by its action on a basis.

Let  β1 , … , βn be a basis of a vector space V ,

and w1 , …, wn are (perhaps not distinct) elements of a vector space W .

Then there exists a unique homomorphism h : V →W s.t. h(βk ) = wk k

Proof:

Define h : V →W by

Then

→ h is a homomorphism

Let g be another homomorphism s.t. g(βk ) = wk . Then

→ h is unique

specifies a homomorphism h: R2 → R2

Definition 1.11: Linear Transformation

A linear map from a space into itself t : V → V is a linear transformation.

Remark 1.12:

Some authors treat ‘linear transformation’ as a synonym for ‘homomorphism’.

Example 1.13: Projection P:R2 → R2

is a linear transformation.

Example 1.14: Derivative Map d /dx: Pn→ Pn

is a linear transformation.

Example 1.15: Transpose Map

is a linear transformation of M22.

It’s actually an automorphism.

Lemma 1.16: L(V,W)

For vector spaces V and W, the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W. It is denoted L(V,W).

Proof: Straightforward (see Hefferon, p.190)

1. Stating that a function is ‘linear’ is different than stating that its graph is a line.

(a) The function f1 : R →R given by f1(x) = 2x 1 has a graph that is a line. Show that it is not a linear function.

(b) The function f2 : R2 →R given by

does not have a graph that is a line. Show that it is a linear function.

2. Consider this transformation of R2.

What is the image under this map of this ellipse.

Lemma 2.1:

Let h: V→W be a homomorphism between vector spaces.

Let S be subspace of V. Then h(S) is a subspace of W. So is h(V) .

Proof: s1 , s2 V and a, b  R,

QED

Definition 2.2: Rangespace and Rank

The rangespaceof a homomorphism h: V → W is

R(h) = h(V ) = { h(v) | v  V }

dim[ R(h) ] = rank of h

Example 2.3: d/dx: P3 → P3

Rank d/dx = 3

Example 2.4: Homomorphism

h: M22 → P3 by

Rank h = 2

Homomorphism: Many-to one map

h: V → W

Inverse image

Example 2.5: Projection π: R3 → R2 by

= Vertical line

Example 2.6: Homomorphism h: R2 → R1by

= Line with slope 1

Isomorphism i: V n → W n V is the same as W

Homomorphism h: V n → W m V is like W

Example 2.7: Projection π: R3 → R2 R3 is like R2

Example 2.8: Homomorphism h: R2 → R1by

Example 2.9: Homomorphism h: R3 → R2by

Range is diagonal line in x-y plane.

Inverse image sets are planes perpendicular to the x-axis.

Lemma 2.10:

Let h: V → W be a homomorphism.

If S is a subspace of h(V), then h1(S) is a subspace of V.

In particular, h1({0W}) is a subspace of V.

Proof: Straightforward (see Hefferon p.188 )

Definition 2.11: Nullspace or Kernel

The nullspace or kernelof a linear map h: V → W is the inverse image of 0W

N(h) = h1(0W) = { v V | h(v) = 0W }

dim N (h) = nullity

Example 2.12: d/dx: P3 → P3 by

Example 2.13: h: M22 → P3 by

Theorem 2.14:

h: V → W  rank(h) + N(h) = dim V

Proof: Show BV \ BN is a basis for BR (see Hefferon p.189)

Example 2.15: Homomorphism h: R3 → R4by

Rank h = 2

Nullity h = 1

Example 2.16: t: R → Rby x 4x

Rank t = 1

R(t) = R

N(t) = 0

Nullity t = 0

Let h: V → W be a homomorphism.

rank h  dim V

rank h = dim V nullity h = 0 (isomorphism if onto)

Lemma 2.18: Homomorphism preserves Linear Dependency

Under a linear map, the image of a L.D. set is L.D.

Proof: Let h: V → W be a linear map.

with some ck 0

with some ck 0

A linear map that is 1-1 is nonsingular. (1-1 map preserves L.I.)

Example 2.20: Nonsingular h: R2 → R3by

gives a correspondence between R2 and the xy-plane inside of R3.

Theorem 2.21:

In an n-D vector space V , the following are equivalent statements about a linear map h: V → W.

(1) h is nonsingular, that is, 1-1

(2) h has a linear inverse

(3) N(h) = { 0 }, that is, nullity(h) = 0

(4) rank(h) = n

(5) if  β1 , … , βn  is a basis for V

then  h(β1 ), … , h(βn ) is a basis for R(h)

Proof: See Hefferon, p.191

1. For the homomorphism h: P3 →P3 given by

Find the followings:

(a) N(h) (b) h 1( 2  x3 ) (c) h 1( 1+ x2 )

2. For the map f : R2 →R given by

sketch these inverse image sets: f 1(3), f 1(0), and f 1(1).

3. Prove that the image of a span equals the span of the images. That is, where h: V → W is linear, prove that if S is a subset of V then h([S]) = [h(S)].