LECTURE 10
Linear Transformations
1. Functions Between Sets
Although I think it’s safe to assume that everybody here is familiar with the utility of functions, it may not
be the case that everybody keeps in mind the generality of this concept. So let me take a few minutes to
remind us of what a function is in set theoretical terms.
Given two sets A and B a function f : A ! B is a rule that links each element of A to a corresponding
element of B. This situation is often represented pictorially as
It is common to say that a function f : A ! B is a map from A to B. We also write a 7 ! f (a) and say
that an element a is mapped to the element f (a) of B. If f : A ! B is a function from A to B
the set A is called the domain of f
the set B is called the codomain of f
the set image (f ) := fb 2 B j b = f (a) for some a 2 Ag is called the image of f
If b 2 B, the set f 1 (b) = fa 2 A j f (a) = bg is called the pullback (or …ber) of b.
It is important to note that if f : A ! B is a function from A to B, then for each element a of A there is
exactly one element f (a) of B. On the other hand,
It is not necessarily true that to each element b 2 B there is an a 2 A such that b = f (a). If it does
happen that for each b 2 B there is an a 2 A such that b = f (a), then the function f : A ! B is
said to be surjective or onto. Equivalently, f : A ! B is surjective if image (f ) = B.
It is not necessarily true that to each element b 2 B rhere is unique element a such that b = f (a).
If
whenever f (a) = f (a0 )
we have a = a0
then the function f : A ! B is called injective or into (or one-to-one, however, this latter
terminology sometimes causes confusion with the notion of a one-to-one correspondence).
A function that both injective and surjective is said to be bijective.
1
2. LINEAR TRANSFORM ATIO NS
2
Theorem 10.1. If f : A ! B is a function that is both surjective and injective, then there exists a function
f 1 : B ! A with the properties that
f
1
f (a) = a
;
8a2A
and
f
f
1
(b) = b
;
8b2B :
Remark 10.2. Although the notation is the same, a pullback f 1 (b) of a non-injective function from A
to B is not the same thing as an inverse function; because a function from B to A would have to have a
unique value for each point in its domain. But sometimes f 1 (b) = fg (e.g, which can happen when f is
not surjective) and sometimes f 1 (b) = fa; a0 ; : : :g which can happen when f is not injective. On the other
hand, when f : A ! B is both injective and surjective, then f 1 can be identi…ed as the inverse function
of A; since in this situation for each b 2 B there will be exactly one element in f 1 (b)
the pullback of b by f = fag
Examples 10.3.
()
the value of the inverse function f
1
at b is a
(1) f : R ! R : x 7 ! x2 is neither surjective nor injective. For
1 6= x2
for any x 2 R
f (2) = f ( 2) = 4
but 2 6= 2 :
(2) By restricting the codomain of the function of f in Example 1 to be R 0 , we can obtain a surjective
function fe : R ! R 0 : x 7 ! x2 :
(3) By restricting the domain of the function of f in Example 1 to be R 0 , we can obtain a injective
function f : R 0 ! R : x 7 ! x2 :
(4) By restricting both the domain and the codomain of the function of f in Example 1 to be R 0 ,
we can obtain a bijective function fe : R ! R 0 : x 7 ! x2 :
2. Linear Transformations
Concentrating as we do in this course on vector spaces, it is natural for us to consider functions between
vector spaces. Surprizingly, it will turn out that the most useful functions between vector spaces are also
practically the most simple functions. We have already seen nice properties of subsets of a vector space
V which are in some sense naturally compatible with the vector operations of scalar multiplication and
vector addition (for a subset of V that closed under these operations is a subspace of V ). It turns out that
requiring functions to be compatible with scalar multiplication and vector addition leads to a particularly
useful class of functions.
Definition 10.4. Let f : V ! W be a function between two vector spaces over a …eld F. We say that f is
a linear transformation if
(i) f ( v) = f (v) for all v 2 V and all 2 F
(ii) f (v + u) = f (v) + f (u) for all v; u 2 V .
;
Remark 10.5. One can check both these conditions simultaneously by checking
f ( v + u) = f (v) + f (u)
for all ;
2 F and for all v; u 2 V
:
Remark 10.6. Any linear function f : x ! ax + b can be regarded as a function that maps R1 to R1 .
However, such a linear function does not de…ne a linear transformation. For example, suppose f (x) = 2x+3.
Then
f (3x) = 6x + 3 6= 3f (x) = 6x + 9
Also
f (x + y) = 2 (x + y) + 3 6= 2x + 2y + 6 = f (x) + f (y)
Upon closer inspection one …nds the source of the discrepancy between a linear function and a linear
transformation is possible occurence of a constant term in a linear function. If on the other hand, one were
to use only linear functions without constant terms, then you’d always get a linear transformation.
3. KERNEL AND IM AGE OF LINEAR TRANSFORM ATION
3
In fact, suppose f : Fm ! Fn is a function beween two standard 1 vector spaces over F. Write
f ([x1 ; : : : ; xm ]) = [f1 (x1 ; : : : ; xm ) ; f2 (x1 ; : : : ; xm ) ; : : : ; fn (x1 ; : : : ; xm )]
Then f will be a linear transformation between Fm and Fn if and only each each of the coordinate functions
fi (x1 ; : : : ; xm ) is a linear function without constant term of its arguments; i.e;, for each i from 1 to n and
each j from 1 to m, there are elements ij 2 F such that
(*)
fi (x1 ; : : : ; xm ) =
i1 x1
+
i2 x2
+
+
Notice that the right hand side of (*) looks an awful lot
2
0
1
x1
:::
11
1m
6 ..
B ..
C
.
.
.
.
=@ .
.
. A ; x=4 .
xm
n1
nm
Then the equations in (*) amount to
f (x) =
im xm
;
i = 1; : : : ; n
like some sort matrix multiplication. Indeed, if set
3
2
3
f1 (x1 ; : : : ; xm )
7
6
7
..
f (x) = 4
5
5 ;
.
fn (x1 ; : : : ; xm )
x
We will better establish the connection between linear transformations and matrices in the next lecture.
3. Kernel and Image of Linear Transformation
Recall that if f : A ! B is a map between two sets then the image of f is
image (f ) = fb 2 B j b = f (a) for some a 2 Ag
and if b 2 B the pullback of b by f is
f
1
(b) = fa 2 A j b = f (a)g
Note that the de…nition of the image of f constructs a certain subset of the codomain B while the de…ntion
of the pullback of an element b in the codomain constructs a certain subset of the domain. These two
constructions especially important to linear transformations.
Theorem 10.7. Let T : V ! W be a linear transformation between two vector spaces over a …eld F. Then
the image of T is a subspace of the codomain W .
Proof. Suppose w1 and w2 lie in image of T . We want to show that if ; 2 F then w1 + w2 lies
in the image of T (and so the image is closed under scalar multiplication and vector addition). Since
w1 ; w2 2 image (T )
w1
= T (v1 )
w2
= T (v2 )
for some v1 2 V
for some v2 2 V
But then
w1 + w2
=
T (v1 ) + T (v2 )
= T ( v1 ) + T ( v2 )
= T ( v1 + v2 )
by property (i) of a linear transformation
by propert (ii) of a lnear transformation
and so w1 + w2 2 image (T ).
Theorem 10.8. Let T : V ! W be a linear transformation. Then the pullback of 0W 2 W by T is a
subspace of the domain V .
1 Fm will be regarded as a standard vector space ove r F in the same sense as Rm is regarded as the standard m-dimensional
vector space in Math 3013.
3. KERNEL AND IM AGE OF LINEAR TRANSFORM ATION
Proof. We …rst note that T
1
4
(0W ) is non-empty. Indeed, by property (i) of a linear transformation
T (0V ) = T (0F v) = 0F T (v) = 0W
So 0V 2 T
1
(0W ). Now suppose that ;
T ( v1 + v2 )
2 F and v1 ; v2 2 T
= T ( v1 ) + T ( v2 )
=
T (v1 ) + T (v2 )
=
0W +
0W
= 0W
Thus, v1 + v2 2 T
1
1
(0W ). Then
by property (ii) of a linear transformation
by property (i) of a linear transformation
since v1 ; v2 2 T
1
(0W )
(0W ).
These two subspaces image (T ) and T
them:
1
(0) are so important in linear algebras, we have special names for
Definition 10.9. Let T : V ! W be a linear transformation between two vector spaces. The subspace of
W given by
T (V ) = range (T ) = fw 2 W j w = T (v) for some v 2 V g
is called the range of T . The subspace of V given by
ker (T ) = T
is called the kernel of T .
1
(0W ) = fv 2 V j T (v) = 0W g