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8.1 In Section 4.2 we defined linear transformations from to . In this
section we shall extend this idea by defining the more general concept of a
GENERAL LINEAR linear transformation from one vector space to another.
TRANSFORMATIONS
Definitions and Terminology
Recall that a linear transformation from to was first defined as a function
for which the equations relating , ,…, and , ,…, are linear. Subsequently, we showed that a transformation
is linear if and only if the two relationships
hold for all vectors u and v in and every scalar c (see Theorem 4.3.2). We shall use these properties as the starting point
for general linear transformations.
DEFINITION
If is a function from a vector space V into a vector space W, then T is called a linear transformation from V
to W if, for all vectors u and v in V and all scalars c,
(a)
(b)
In the special case where , the linear transformation is called a linear operator on V.
EXAMPLE 1 Matrix Transformations
Because the preceding definition of a linear transformation was based on Theorem 4.3.2, linear transformations from to
, as defined in Section 4.2, are linear transformations under this more general definition as well. We shall call linear
transformations from to matrix transformations, since they can be carried out by matrix multiplication.
EXAMPLE 2 The Zero Transformation
