ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Let L be a linear mapping from the vectors pace V to the vector space W .
Then L is a homomorphism wrt the pair (+, +).
Let Λ denote multiplication with the scalar λ, that is Λ(x) = λx for any vector
x. Then Λ is a unary operator and L is a homomorphism wrt the pair (Λ, Λ)
Example 4: Presenting a linear mapping by a matrix is homomorphic (see nr
8)
If L is a linear mapping from the linear space V to the linear space W and
bases are chosen, then there exists a unique matrix A such that if w = L(v)
then y = Ax where x are the coordinates of v and y are the coordinates of w.
We say that A represents L (or is associated with or belongs to L).
The mapping that sends L to A is a homomorphism wrt the pair of binary operations
(◦, ·) consisting of composition of maps and multiplication of matrices.
If we restrict to invertible mappings and invertible matrices then it is also
a homomorphism wrt the pair of unary operations (inversion of mappings,
inversion of matrices).
Example 5: Associating a Möbius transformation with a matrix is homomorphic
(see nr 8)
(
a
To any invertible complex matrix
c
hA given by
)
b
we associate the complex mapping
d
hA(z) =
az + b
cz + d ,
The mapping that takes A to hA is a homomorphism wrt to the pair (·, ◦) of
matrix multiplication and composition of mappings.
Example 6: The dependence of the power on the exponent is a homomorphy
(see nr 8)
Let x be a real number. Then the mapping N ∋ n ↦→ x n ∈ R is a homomorphism
wrt the pair (+, ·).
Example 7: The dependence of the matrix power on the exponent is a homomorphy
(see nr 8)
Let A be a square matrix. Then the mapping N ∋ n ↦→ A n is a homomorphism
wrt the pair (+, ·).
Example 8: Homomorphisms for unary operations (see nr 8)
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