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Let V and W be any two vector spaces. The mapping such that for every v in V is a linear
transformation called the zero transformation. To see that T is linear, observe that
Therefore,
EXAMPLE 3 The Identity Operator defined by is called the identity operator on V. The verification
Let V be any vector space. The mapping
that I is linear is left for the reader.
EXAMPLE 4 Dilation and Contraction Operators defined by
Let V be any vector space and k any fixed scalar. We leave it as an exercise to check that the function
is a linear operator on V. This linear operator is called a dilation of V with factor k if and is called a contraction of V
with factor k if . Geometrically, the dilation “stretches” each vector in V by a factor of k, and the contraction of V
“compresses” each vector by a factor of k (Figure 8.1.1).
Figure 8.1.1
