Page 285 - 35Linear Algebra
P. 285
16
Kernel, Range, Nullity, Rank
Given a linear transformation
L: V → W ,
we often want to know if it has an inverse, i.e., if there exists a linear trans-
formation
M : W → V
such that for any vector v ∈ V , we have
MLv = v ,
and for any vector w ∈ W, we have
LMw = w .
A linear transformation is a special kind of function from one vector space to
another. So before we discuss which linear transformations have inverses, let
us first discuss inverses of arbitrary functions. When we later specialize to
linear transformations, we’ll also find some nice ways of creating subspaces.
Let f : S → T be a function from a set S to a set T. Recall that S is
called the domain of f, T is called the codomain or target of f. We now
formally introduce a term that should be familar to you from many previous
courses.
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