Page 77 - Advanced Linear Algebra
P. 77
Linear Transformations 61
specifying the values of for all arbitrarily # 8# and extending to by
=
linearity, that is,
² # b Äb # ³ ~ # bÄb #
This process defines a unique linear transformation, that is, if Á B = ² Á > ³
satisfy #~ # for all # 8 then ~ .
Proof. The crucial point is that the extension by linearity is well-defined, since
each vector in = has an essentially unique representation as a linear
combination of a finite number of vectors in . We leave the details to the
8
reader.
B
=
Note that if ²= Á > ³ and if is a subspace of , then the restriction O : of
:
to is a linear transformation from to > .
:
:
The Kernel and Image of a Linear Transformation
There are two very important vector spaces associated with a linear
transformation from to > .
=
B
Definition Let ²= Á > ³ . The subspace
ker² ³ ~¸#= #~ ¹
is called the kernel of and the subspace
im² ³ ~¸ ##= ¹
is called the image of . The dimension of ker²³ is called the nullity of and is
denoted by null²³ . The dimension of im ²³ is called the rank of and is
denoted by rk²³ .
=
It is routine to show that ker²³ is a subspace of and im ²³ is a subspace of
> . Moreover, we have the following.
B
Theorem 2.3 Let ²= Á > ³ . Then
1 ) is surjective if and only if im ²³ ~ >
2 ) is injective if and only if ker² ³ ~ ¸ ¹
Proof. The first statement is merely a restatement of the definition of
surjectivity. To see the validity of the second statement, observe that
"~ #¯ ²" c #³~ ¯" c # ker ² ³
Hence, if ker² ³ ~¸ ¹ , then "~ #¯"~# , which shows that is injective.
Conversely, if is injective and " ker ² ³ , then "~ and so " ~ . This
shows that ker² ³ ~ ¸ ¹ .
