Maths - Morphism Examples

Here we look at some examples to see if they are morphisms, that is, does:

φ(v1) o φ(v2) = φ(v1 o v2)

where:

isomorphiism

Example 1 vector addition with mapping = shift of origin

vector addition shift

With mapping φ between vector additions of fixed offset Vs so each vector is transformed by:

φ(V) = V + Vs

This is not an isomorphism since:

(V1 + Vs) + (V2 + Vs) ≠ (V1 + V2) + Vs

however:

(V1 + Vs) + (V2) = (V1 + V2) + Vs

so if we treat V2 as an offset and don't transform it then the equation matches.

Example 2 vector addition with mapping = scaling

vector addition scale

With mapping φ between vector spaces of scalar multiplication by M so each vector is transformed by:

φ(V) = M*V

This is an isomorphism since:

V1*M + V2*M = (V1 + V2) *M

Example 3 multiplication with mapping = scaling

vector multipy scale

With mapping φ between vector additions of fixed scalar multiplication by M so each vector is transformed by:

φ(V) = M*V

This is not an isomorphism since:

V1*M * V2*M ≠ (V1 * V2) *M

So what mapping φ can we use with multipication to form an isomrphism? A common form is the 'sandwich product':

φ(v) = q v q-1

I have used q here to indicate a quaternion because the various types of vector product don't really have the correct properties. Use of the sandwich product in this way is discussed on this page.

so

φ(q1*q2) = q q1 q2 q-1
= q q1 q-1 q q2 q-1
= φ(q1)*φ(q2)

(since q-1 q =1)

So we can see that this sandwich product (conjugacy) has the right properties for an isomorphism


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