Linear transformation, linear mapping. Operations, sum, product. Algebra of linear operators. Invertible operators.

 As with many concepts of modern mathematics the concept of a linear transformation is very abstract. Everything has been stripped away from it except that which is most fundamental and essential. All that remains is a skeleton consisting of a very simple concept subject to certain axiomatic conditions. The simple concept is that of a function that assigns objects in one set (the co-domain) to objects in another set (the domain) in a certain way.

Def. Linear Transformation. Let V and W be vector spaces over the field F. A linear transformation from V into W is a function T from V into W such that

1) T(x+y) = Tx + Ty

and

2) T(cx) = cTx

for all x and y in V and all scalars c in F.

When one derives theorems about linear transformations they are deduced directly from this axiomatic definition. Nothing more is assumed. One proceeds from axiomatic assumptions to conclusions in the same way as one does in Euclidean geometry.

By combining the above two conditions of linearity we obtain the following basic property of linear transformations:

Basic property. If T: V W is a linear transformation over a field F, then for any scalars a_i in F and vectors v_i in V

            T(a₁v₁ + a₂v₂ + ......... + a_nv_n) = a₁T(v₁) + a₂T(v₂) + ............... + a₂T(v₂)

Sums of linear transformations.

Suppose we have more than one linear transformation defined on a vector space. Suppose we have two linear transformations, T and U, defined on some space V and mapping into some space W. The sum T + U of the two transformations has no natural meaning. The idea of adding two such transformations makes no sense. But investigators in the field of mathematics have had their reasons for wanting to be able to work with the sums of such transformations. So they have become inventive. Just as they have done in many other cases in mathematics (e.g. creation of negative numbers, imaginary numbers, etc.) they have defined and invented to suit their purposes. They have defined a meaning for the sum of two such transformations. By this defined meaning:

3) (T+U)x = Tx + Ux

for any vector x in vector space V. In the same way they have also defined a meaning for the idea of a scalar times a linear transformation, that is for the product "cT" for the scalar c and transformation T. Without some defined meaning this also has no natural meaning and makes no sense. Their defined meaning is:

4) (cT)x = c(Tx)

for any scalar c and any x in vector space V. With these two definitions they have created a system which is itself a vector space of transformations, i.e. the linear transformations themselves constitute a vector space. More precisely, let V and W be vector spaces over the field F. Then the set of all linear transformations from V into W, with addition and scalar multiplication as defined above, is a vector space over F

We could ask the following question: What is the benefit of doing a thing like we have just described -- defining meanings for things which have no natural meaning in order to create a system that meets the requirements of a linear space? What kind of game is that? Why do that? Well, although the ideas of the sum of two transformations and the product of a scalar times a transformation make no sense for the abstract concept of a linear transformation as we have defined it, these things do make sense for a lot of important actual linear transformations. For example, in the linear transformations effected by operators such as matrices or integral operators, the ideas of the sums of two operators and the product of a scalar times an operator does make sense and gives transformations which conform to the above definitions 3) and 4) meaning. And in these cases, by assuming one can add two operators and multiply them by scalars in the natural way, one does naturally obtain a vector space of operators. So the problem was one that came up in the process of creating the very abstract, axiomatic definition of the linear transformation. In specific, actual cases of linear transformations these operations made sense but in the proposed axiomatic definition they didn't so they solved the problem by defining meanings for the operations.

We shall now change terminology and use the term linear mapping instead of linear transformation.

Operations with linear mappings

Addition of linear mappings. Let P: U V and Q: U V be linear mappings over a field F. Then the sum P + Q is defined as

            (P + Q)x = Px + Qx

for any vector x in U.

Scalar product. For a mapping P: U V over a field F, the scalar product cP is defined as

            (cP)x = c(Px)

for any scalar c in F.

Theorem 1. Let V and W be vector spaces over a field F. Then the set of all possible linear mappings (or linear transformations, functions or operators) from V into W with the above defined operations of addition and scalar multiplication form a vector space over F. This vector space is the space Hom(V,W). See

Hom(V,W). Vector space of all mxn matrices

Vector space Hom(V,W)

Theorem 2. Let U and V be of finite dimension with dim U = m and dim V = n. Then dim Hom(U, V) = mn.

Products of linear mappings. Let U, V and W be vector spaces over the same field F. Let P: U V and Q: V W be linear mappings from U into V and V into W respectively. See Fig. 1. Then the product QP is defined to be the composition function Q P given by

            (Q P)v = Q(Pv)

See Functions, Mappings, etc. – Product function

Theorem 3. If functions P and Q are linear, then the product QP is also linear.

Proof

Theorem 4. Let U, V and W be vector spaces over the same field F. Let P and P' be linear mappings from U into V and Q and Q' be linear mappings from V into W and c be any scalar in F. Then:

1)        Q(P + P') = QP + QP'

2)        (Q + Q')P = QP + Q'P

3)        c(QP) = (cQ)P = Q(cP) 

Algebra of linear operators

Def. Algebra over a field. An algebra A over a field F is a vector space over F in which an operation of multiplication is defined satisfying, for every P, Q, R in A and every c in F,

1)        P(Q + R) = PQ + PR

2)        (Q + R)P = QP + RP

3)        c(QP) = (cQ)P = Q(cP)

If the associative law

4)        (PQ)R = P(QR)

also holds, the algebra is said to be associative.

Linear space Hom(V,V). Let V be a vector space over a field F. Let us denote by A(V) the space Hom(V,V) of all linear mappings (or linear transformations or operators) of V into itself. If V is of dimension n, then A(V) will be of dimension n². For any two mappings P, Q ε A(V) the composition product mapping PQ also exists and is also a mapping from V into itself i.e. PQ ε A(V). Thus multiplication is defined in A(V).

Theorem 5. Let V be a vector space over a field F. Then A(V) = Hom(V,V) is an associative algebra over F with respect to composition of mappings. It is frequently called the algebra of linear operators on V.

Def. Invertible operator. A linear operator P: V V is said to be invertible if it has an inverse i.e. if there exists P ^-1 ε A(V) such that PP ^-1 = P ^-1 P = I.

Theorem 6. A linear operator P: V V on a vector space of finite dimension is invertible if and only if it is nonsingular.

References

Lipschutz. Linear Algebra. pp. 128-129