Definitions and Terminology

We start with a general definition of the homomorphism of algebraic structures.

Suppose we have two sets, each with a binary operation: \(\,(A,\,\bot\,)\ \) and \(\,(A',\,\circ\,).\)

A map \(\ f:\ A\,\rightarrow\,A'\ \,\) is \(\,\) a \(\,\) homomorphism, when it preserves the operations in the sense that the image of a composition of two elements of \(\,A\,\) is equal to the composition (in the set \(\,A'\,\)) of their images:

(1)\[f(a_1\,\bot\;a_2)\ =\ f(a_1)\,\circ\,f(a_2)\,, \qquad\forall\ a_1,a_2\in A\,.\]

In the case of several internal operations defined in the two sets, the condition (1) should be fulfilled for each pair of corresponding operations. Homomorphisms and isomorphisms (i.e. bijective homomorphisms) of groups, rings and fields are defined just that way.

A vector space contains the external operation of multiplying vectors by scalars. The properly modified definition reads as follows.

Definition.

Let \(\ V\ \) and \(\ W\ \) be two vector spaces over (the same) field \(\,K.\)

The map \(\ \,F:\ V\rightarrow W\ \,\) is the \(\,\) homomorphism \(\,\) \(\,\) (linear transformation) \(\,\)

of the space \(\ V\ \) into the space \(\ W,\ \ \) when it is simultaneously:

1. additive: \(\quad F(v_1+v_2)\ =\ F(v_1)+F(v_2)\,,\qquad\forall\ \ v_1,v_2\in V\ \)

   (the image of a sum of two vectors equals the sum of their images)
2. homogeneous: \(\quad F(a\,v)\,=\,a\,F(v)\,, \qquad\forall\ a\in K,\ \forall\ v\in V\)

   (multiplying a vector \(\,v\,\) by a number \(\,\alpha\,\) multiplies by \(\,\alpha\,\) the image of \(\,v\)).

The conditions of additivity and homogeneity can be encapsulated in the single equation:

(2)\[F(a_1\,v_1+a_1\,v_2)\ \,=\ \,a_1\,F(v_1)\,+\,a_2\,F(v_2)\,,\qquad \forall\ \ a_1,a_2\in K,\ \forall\ \ v_1,v_2\in V\,.\]

(the image of a linear combination of two vectors equals the same combination of their images).

By induction, one may generalize Eq. (2) to the case of a linear combination of any finite number of vectors.

The addition and scalar multiplication of mappings being defined in a natural way, the set of all linear transformations of the vector space \(\ V\ \) into the space \(\ W\ \) turns out to be a vector space itself. To deal with this matter in details, we use the following notation:

\(\text{Map}(V,W)\ \) - \(\,\) the set of all maps from the vector space \(\,V\ \) to the vector space \(\,W;\)

\(\text{Hom}(V,W)\ \) - \(\,\) the set of all linear maps (homomorphisms) from \(\,V\ \) into \(\,W.\)

Verifying the postulates in the definition of vector space, one may prove

Lemma 1. \(\\\)

If \(\ V\ \) and \(\ W\ \) are vector spaces over the field \(\,K,\ \) then the set \(\ \text{Map}(V,W)\ \) \(\\\) with the operations of addition and scalar multiplication:

(3)\[\begin{split}\begin{array}{lcl} (F_1+F_2)(v)\ :\,=\ F_1(v)\,+\,F_2(v) & \qquad & \forall\ F_1,F_2,F\in \text{Map}(V,W)\,, \\ (a\,F)(v)\ :\,=\ a\,F(v) & \qquad & \forall\ \,a\in K,\ \ \forall\ v\in V\,, \end{array}\end{split}\]

is also a vector space over the field \(\,K.\)

(the statement remains true for a set \(\ \text{Map}(X,W),\ \) where \(\ X\ \) is any given set).

It’s also easy to justify

Lemma 2. \(\\\)

If \(\ V\ \) and \(\ W\ \) are vector spaces over the field \(\,K,\ \) then the set \(\text{Hom}(V,W)\ \) \(\\\) is closed with respect to the addition and the scalar multiplication of mappings. \(\\\) Namely, for any \(\ F_1,F_2\in\text{Map}(V,W),\ a\in K:\)

\[F_1,\,F_2\in \text{Hom}(V,W)\quad\Rightarrow\quad \left[\ (F_1+F_2)\in\text{Hom}(V,W) \ \ \wedge\ \ (a\,F_1)\in\text{Hom}(V,W)\ \right]\,.\]

Therefore, the criterion for a subset of a vector space to be a subspace leads to

Theorem 1. \(\\\)

Let \(\ V\ \) and \(\ W\ \) be vector spaces over a field \(\,K.\ \) Then the set \(\text{Hom}(V,W)\ \) of linear transformations of the space \(\ V\ \) into \(\ W\ \) is also a vector space over \(\,K;\ \) moreover, it is a subspace of the vector space \(\text{Map}(V,W)\ \) of all transformations of \(\ V\ \) into \(\ W:\)

\[\text{Hom}(V,W)\ <\ \text{Map}(V,W)\,.\]

A bijective linear transformation of a space \(\ V\ \) into a space \(\ W\ \) is named an \(\,\) isomorphism \(\,\) of these vector spaces. The collection of all such isomorphisms is denoted by \(\,\text{Iso}(V,W).\ \) \(\\\) When \(\ \text{Iso}(V,W)\ne\emptyset,\ \) one says that the vector spaces \(\ V\ \) and \(\ W\ \) are \(\,\) isomorphic: \(\ V\simeq W.\)

A linear transformation of the vector space \(\ V\ \) into itself is called an \(\,\) endomorphism \(\,\) or a \(\,\) linear operator. \(\,\) Isomorphism of a space onto itself, i.e. a bijective endomorphism, is called \(\,\) automorphism. \(\,\) The collections of such transformations are denoted by

\[\text{End}(V)\ :\,=\ \text{Hom}(V,V)\,,\qquad \text{Aut}(V)\ :\,=\ \text{Iso}(V,V)\,.\]

The relations between these notions can be charted by the following scheme, in which the horizontal right-arrows symbolize the condition of bijectivity, whereas the vertical down-arrows designate the substitution \(\ W=V:\)

\[\begin{split}\left.\begin{array}{ccc} \text{homo-} & \longrightarrow & \text{iso-} \\ \\ \downarrow & & \downarrow \\ \\ \text{endo-} & \longrightarrow & \text{auto-} \end{array}\quad\right\} \quad\text{-morphism}\,.\end{split}\]

In addition to be added and scalar multiplied (Eqs. (3)), \(\,\) the linear operators in \(\ \text{End}(V)\ \) can be composed according to the rule

\[(F\circ G)(v)\ \,:\,=\ \, F\,[\,G(v)\,]\,,\qquad F,\,G\in\text{End}(V),\ \ \forall\ v\in V\,.\]

Checking up the postulates in the definition of algebra, one may validate

Theorem 2. \(\,\)

Let \(\ V\ \) be a linear space over the field \(\ K.\) Then the set \(\ \text{End}(V)\ \) of all linear operators defined on \(\ V,\ \) together with the operations of adding, scalar multiplying and composing the operators, \(\,\) is a non-commutative algebra over the field \(\ K.\)

If \(\ V\ \) is a vector space over the field \(\ K,\ \) then the linear transformation \(\ f:\ V\rightarrow K,\ \) where the field \(\ K\equiv K^1\ \) is considered as the one-dimensional vector space over \(\,K,\ \) is called \(\,\) the \(\,\) linear functional. The set \(\ V^\ast :\,=\ \text{Hom}(V,K)\ \) of all linear functionals defined on the space \(\, V,\ \,\) is the \(\,\) dual space \(\,\) of \(\ \,V.\ \)