Kernel and Range
Definition of one-to-one transformation:
A linear transformation
V,
v1  v2
L :V  W is said to be one-to-one if for all v1 , v2 in
implies that
Lv1   Lv2 
(or
Lv1   Lv2 
implies
v1  v2 ).
Example:
  x  x  y
L     
 . Is L one-to-one?
y
x

y

   
[solution:]
Let
x 
x 
v1   1 , v2   2  . Then,
 y1 
 y2 
 x  y1   x 2  y 2 
Lv1   Lv 2    1
   x  y   x1  y1  x2  y 2
x

y
1
2
 1
 2
x1  y1  x2  y 2
 2 x1  2 x2  x1  x2  y1  y 2 .
x 
x 
 v1   1    2   v 2
 y1 
 y2 
.
Therefore, L is one-to-one.
Definition of kernel:
Let
L :V  W be a linear transformation. The kernel of L,
1
kerL , is
v
the subset of V consisting of all vectors
Lv   0 .
such that
Example:
Let
  u1  
1 2 3  u1 
  
Lu   L u 2    Au  4 5 6 u 2 
.
 u  
7 8 9 u 3 
  3 
What is
kerL ?
[solution:]
kerL
is the set consisting of all vectors
 u1 
u  u 2 
u 3 
such that
Lu   Au  0 .
That is,
kerL
is the solution space of
Au  0 .
Important result:
Let L :V  W be a linear transformation. Then,
subspace of V.
[proof:]
For any
v1, v2  kerL , Lv   Lv   0 . Then,
1
2
2
kerL
is a
1.
Lv1  v2   Lv1   Lv2   0  0  0  v1  v2  kerL .
2.
Lkv1   kLv1   k 0  0, k  R  kv1  kerL .
By 1, 2,
kerL
is a subspace of V.
Important result:
A linear transformation L :V  W is one-to-one if and only if
ker L  0
[proof:]
:
Since L is one-to-one, L0  0 implies there is only one vector
0 such that L0  0 .
: kerL  0. Suppose Lu   Lv . Then,
Lu  v  Lu   Lv  0 . Since ker L   0, that implies
u  v  0  u  v . Therefore, L is one-to-one.
Important result:
n
n
Let L : R  R be a linear transformation defined by Lx  Ax ,
where A is a
equivalent:
n n

ker L  0.

Ax  0
matrix. Then, the following conditions are
has only the trivial solution.
3

Ax  b

Lx  Ax
has a unique solution for every
b.
is one-to-one.
Definition of range:
Let
L :V  W be a linear transformation. The range of L,
rangeL ,
is the subset of W consisting of all vectors that are images of vectors in V (i.e.,
w rangeL , there exists
for any
If
rangeL  W
v in V such that
Lv   w ).
, L is said to be onto.
Important result:
Let L :V  W be a linear transformation. Then,
rangeL
is a
subspace of V.
[proof:]
Since
L0  0 , 0  rangeL .
empty set. For any
such that
1.
Thus,
rangeL
is not a
w1, w2  rangeL , there exist v1 , v2
in V
Lv1   w1 , Lv2   w2 . Then,
Lv1  v2   Lv1   Lv2   w1  w2 .
Then, w1  w2  rangeL
since v1  v2 V .
2.
Lkv1   kLv1   kw1, k  R
. Then,
kv1 V
By 1, 2,
rangeL
is a subspace of V.
4
kw1  rangeL since
Example:
Let
L : R3  R3 ,
1 0 1  x1 
Lx   Ax  1 1 2  x 2  .
2 1 3  x3 
(a) Is L onto?
(b) What is range L?
[solution:]
a 
w  b   R 3 , there exists x
 c 
(a) Suppose L is onto. Then for any vector
such
that
1 0 1  x1 
1
0
1 a 
Ax  1 1 2  x2   x1 1  x2 1  x3 2  b   w
.
2 1 3  x3 
2
1
3  c 
That is
 1  0  1  
      
  1  , 1  ,  2  
  2  1   3  
      
span R 3 
 1  0  1  
      
  1  , 1  ,  2  
  2  1   3  
      
is a basis for R 3 .
But
1
det  A  1
2
0
1
1
5
1
2  0 .
3
Thus,
 1  0  1  
      
  1  , 1  ,  2  
  2  1   3  
      
is not a basis and it is a contradiction. Therefore, L is not on-to.
(b)
rangeL   the vectors Ax for any x  R 3
 x1 
 x1 
 A x 2   col1  A col 2  A col 3  A x 2 
 x3 
 x3 
 x1col1  A  x 2 col 2  A  x3 col 3  A, x1 , x 2 , x3  R
 spancol1  A, col 2  A, col 3  A
 the column space of A
Note:
As the linear transformation defined by
L : R n  R m , L x   Am n x,
then
and
rangeL  the column space of A
kerL  the null space of A .
Important result:
Let L :V  W be a linear transformation of an n-dimensional vector
space V into a vector space W, then,
dim rangeL  dim kerL  n  dim V  ,
6
where
dim 
is the dimension of some vector space.
Note:
As the linear transformation defined by
L : R n  R m , L x   Am n x,
the above important result is
rank  A  nullity  A  dim column space of A  dim null space of A
 dim rangeL   dim ker L 
 
 dim R n
n
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