Page 122 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 5. Let T be the unique linear operator on C for which
3
T = (1, 0, i), T = (0, 1, 1,) T = (i, 1, 0).
1 2 3
Is T invertible?

7.4 Summary

 The properties of linear transformations are important in understanding the properties of
the vector space.
 The basis vectors play an important part in the study of linear transformations.

 It is also explained that not all transformations are linear.
 A linear transformation T from a vector space V to a vector space W, both over the same
field is a mapping of V onto W such that for all , V and for all a, b, F,

( T
(a b )T ( a  Τ ) b  )
7.5 Keywords

Homomorphism: If every vector of W is in the range of H, H is said to be homomorphism of V
onto W.
Isomorphism: A one-to-one homomorphism H of V onto W is called an isomorphism. If such a
mapping exists, V, and W are said to be isomorphic.
Linear Transformation: If T is a linear transformation of v into w and T is the linear
1 2
transformation of w into z space, then T T is a linear transformation of v into z.
1 2
7.6 Review Questions

1. Let T be a linear transformation on R defined by
3
T(x , x , x ) = (3x , x – x , 2x + x + x )
1 2 3 1 1 2 1 2 3
(a) Is T invertible? If so, find a rule for T like the one which defines T.
–1
(b) Find the value of

2
(T – I) (T – 3I) (x , x , x ).
1 2 3
2. Let C be the complex vector space of 2 × 2 matrices with complex entries. Let
2×2
1 1
B 4 4

and let T be a linear operator on C defined by
2×2
T(A) = BA – AB
for any A C . What is the rank of T?
2×2

3. A transformation T on vector V of a vector space w is defined by
  
T ( ) A V
V

where the given vector A W and ‘x’ means the vector product. Find

2
V
(T 2 A T ) ( ) .

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