Page 116 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
Example 9: In the space V let T , T and T be defined by
2 1 2 3
(x, y)T = (x, 0)
1
(x, y)T = (0, y)
2
(x, y)T = (y, x)
3
All these transformations are linear, now
(x, y)T T = (x, 0)T = (0, 0), so T T = Z
1 2 2 1 2
But T Z and T Z Hence a product of non-zero transformation can be the zero transformation.
1 2
Also
(x, y)T T = (0, y)T = (y, 0)
2 3 3
But
(x, y)T T = (y, x)T = (0, x). Hence
3 2 2
T T T T .
2 3 3 2
So the multiplication of transformation is not commutative. Observe that

(x, y)T T = (x, 0)T = (x, 0) = (x, y)T ,
1 1 1 1
2
so that T = T . Thus there exist idempotent transformation i.e.
1 1
k
T = T
1 1
for integer k, other than I and Z.
Rank and Nullity of a Linear Transformation
Consider a linear transformation from a space v into a space w. The domain of T is the space v and
the range of T is a subset R of w, the set of all images T of the vectors of v:
T
R = { w| = T for some v}
T
Another set associated with any vector space homomorphism T is the Kernel K of the
T
homomorphism, which is defined to be the set of all vectors in v which are mapped into .
K = { v| T = 0 }.
T
To see that K is a subspace of v, let , K , and C F. Then
T T
( + )T = T + T = = ,
so that + K , also (C )T = C( T) = , so c K ,
T T
Thus K is a subspace of v.
T
These two subspaces, R and K , are called respectively the range space of T and the null space of
T T
T.
The range space R of a linear transformation T is the set of all images T w as ranges over v.
T
The rank p (T) of a linear transformation T is the dimension of its range space.
The nullity v (T) of a linear transformation T is the dimension of its null space.

Consider an n dimensional vector space v . If T is a linear transformation from v to w, then
n n
P(T) = v(T) = n

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