Page 121 - DMTH502_LINEAR_ALGEBRA
P. 121
Unit 7: Algebra of Linear Transformation
Theorem 8: Let T be a linear transformation on v to w the following statements are equivalent. Notes
n n
1. T is non-singular
2. For all , v , if T = T, then = .
n
3. K = [ ]
T
4. v(T) = 0
5. T is onto, that is, the range of T is w i.e. p(T) = n.
n
6. T maps any basis for v onto a basis for w .
n n
Proof: Let n = dim v = din w. Now
rank (T) + nullity (T) = n
Since T is non-singular if and only if nullity (T) = 0 and rank (T) = n. Therefore T is non-singular
if and only if T(v ) = w . So, if either condition (1) or (2) holds the other is satisfied as well and T
n n
is invertible.
The above equations are also equivalent, there is some basis ( , , ) for v such that (T , T ,
1 2 n 1 2
....,, T ) is basis for w.
n
2
Example 10: Let F be a field and let T be the linear operator on F defined by
T (x , x ) = (x , x , x )
1 2 1 2 1
Then T is non-singular.
Proof: If T is singular than T(x , x ) = 0, means we have
1 2
x + x = 0
1 2
x = 0
1
so the solution is x = 0, x = 0. We also see that T is onto; for let (z , z ) be any vector in F .
2
1 2 1 2
To show that (z , z ) is in the range of T we must find scalars z and z such that
1 2 1 2
x + x = z
1 2 1
x = z
1 2
–1
and the obvious solution is x , = z , x = z – z . This last result gives us an explicit for T , namely
1 2 2 1 2
–1
T (x , x ) = (z , z – z )
1 2 2 1 2
Self Assessment
2
4. If T and U be the linear operator on R defined by
T(x , x ) = (x , x ) and U(x , x ) = (x , 0)
1 2 2 1 1 2 1
give rules like the ones defining T and U for each of the transformations
(i) U + T
(ii) UT
(iii) TU
LOVELY PROFESSIONAL UNIVERSITY 115