Page 295 - 35Linear Algebra

16.2 Image 295

Thus
   
 1 1 
2
L(R ) = span    
,
1
2
0 1
 
Hence, when bases and a linear transformation is are given, people often refer to its
range as the column space of the corresponding matrix.
To find a basis of the range of L, we can start with a basis S = {v 1 , . . . , v n }
n
1
for V . Then the most general input for L is of the form α v 1 + · · · + α v n .
In turn, its most general output looks like
n
1
n
1

L α v 1 + · · · + α v n = α Lv 1 + · · · + α Lv n ∈ span{Lv 1 , . . . Lv n } .
Thus
L(V ) = span L(S) = span{Lv 1 , . . . , Lv n } .
However, the set {Lv 1 , . . . , Lv n } may not be linearly independent; we must
solve
n
1
c Lv 1 + · · · + c Lv n = 0 ,
to determine whether it is. By finding relations amongst the elements of
L(S) = {Lv 1 , . . . , Lv n }, we can discard vectors until a basis is arrived at.
The size of this basis is the dimension of the range of L, which is known as
the rank of L.

Definition The rank of a linear transformation L is the dimension of its
range. The nullity of a linear transformation is the dimension of the kernel.

The notation for these numbers is

null L := dim ker L,

rank L := dim L(V ) = dim ran L.

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