Kernel of Linear Transformation:
Let T: V U be a linear Transformation then
Kernel of Linear Transformation T is denoted by Ker T and defined as
Ker T= { v € V / T(v)=0}
Note:1)
Let T: V − − −→ U be a linear Transformation then Kernel of Linear Transformation is subspace of the vector space V and dimensions of subspace ker T is called as the nullity of T.
Image of Linear Transformation:
Let T: V − − −→ U be a linear Transformation then Image of Linear Transformation T is denoted by Im T and defined as
Im T = { u ∈ U / there exist v ∈ V for which T(v)=u }
Note 2)
Let T: V − − −→ U be a linear Transformation then Im T is
subspace of a vector space U.
Rank of a Linear Transformation:
Rank of a Linear Transformation T is defined to be dimensions of its Image And
rank(T) = dim(Im T )
Rank nullity Theorem:
Let T: V − − −→ U be a linear Transformation then
dim V = rank(T) + nullity(T)
dim V= dim Im T + Dim Ker T
Note:
For a Linear Transformation TX= AX where A is matrix of Transformation
then
Column space of A = Image of T
Basis of Im T = Basis of Column Space of the matrix A.
Example : Verify Rank nullity theorem for following Linear Transformations:
1) T : R^3 − − −→ R^3 defined as T |x| |0|
|y| = |x|
|z| |0|
T |x| |0|
|y|= |x|
|z| |0|
= |0x + 0y + 0z| =| 0 0 0| |x|
|x + 0y + 0z | |1 0 0 | |y|
|0x + 0y + 0z| |0 0 0 | |z|
T(X)= AX
Dim V=3
Rank T= Dim.Im T
Im T = { u ∈ U / there exist v ∈ V for which T(v)=u }
Im T= { u ∈ U / there exist |x| |x|
|y| ∈ R^3 for which T(|y|)=u }
|z| |z|
Im T= { u ∈ U / |0| =u }
|x|
|0|
Im T = { |0| /x is real number}
|x|
|0|
Dim Im T = 1 = rank of T
KerT = { |x| / x,y,z.are real and T |x| |0| }
|y| |y| = |0|
| z| |z| |0|
But T |x| |0|
|y| = |x|
|z| |0|
KerT = { |x| / x,y,z.are real and |0| |0| }
|y| |x| = |0|
|z| |0| |0|
0x+0y+0z=0
X+0y+0z=0
0x+0y+0z=0
|0 0 0| |x| |0|
|1 0 0 | |y| = |0|
|0 0 0 | |z| |0|
Augmented matrix is [A: 0] = [0 0 0 ∶0] interchanging 1st and 2nd equation
[1 0 0 ∶0]
[ 0 0 0 :0]
[A: 0] = [1 0 0 ∶0] which is echelon form and re writing equations x=0
[ 0 0 0 ∶0]
[ 0 0 0: 0]
Two free parameters non trival solutions
Let y= t1 and z=t2
KerT = { [0 ] / t1 and t2 real numbers }
0[t1]
0[t2]
Dim V=3 , Dim of Ker T =2 , Dim Of IM T = 1
Dim V= Dim Im T +Dim KerT
3= 1+2
dim V = rank(T) + nullity(T)
