Page 119 - DMTH502_LINEAR_ALGEBRA

Unit 7: Algebra of Linear Transformation

Theorem 5: Let v be an n-dimensional vector space over the filed F; and let w be an m-dimensional Notes
vector space over F. Then the space L(v, w) is finite dimensional and has dimension mn.
Thus let F be field, v and w vector spaces over F and L the set of all linear transformations from
v into w. The system
£ = {L, F, +, .; ,  }
is a vector space over F.

A special situation arises when we consider the system of all linear transformations of a vector
space v into v itself £ is then a vector space in which the “vectors” are linear mappings of v into
v. So we can define a product S  T of vectors. This vector space over F in which a suitable
product of vectors is defined is called an algebra of linear transformations over F.
A linear algebra £ over the field F is a system

£ = {L, F, +, .; ,  }
which satisfies postulates:
(a) the system {L, F, T, ..., , } is a vector space over F.
(b)  is a binary operation on £, which is closed, associative and bilinear

i.e.
closed T , T £
1 2
Associative T (T T ) = (T T )T
1 2 3 1 2 3
Bilinear T (aT + bT ) = aT T + bT T
1 2 3 1 2 1 3
(aT + bT ) T = aT T + bT T
2 3 1 2 1 3 1
Also the dimension of £ is defined to be its dimension as a vector space.
Theorem 6: Let v, w and z be vector spaces over the field F. Let T be a linear transformation from
v into w and u a linear transformation from w into z. Then the composed function UT defined by
(UT) ( ) = U(T( )) is a linear transformation from v into z.
Proof:

(UT) (C + ) = U[T(C + )]
= U(CT + T )
= C[U(T )] + U (T )
= C(UT)( ) + (UT)( )

we shall be primarily concerned with linear transformation of a vector space into itself. So we
from now on we write ‘T is a linear operator on V’ instead of writing ‘T is a linear transformation
from v into V’.
Definition: If v is a vector space over the field, a linear operator on v is a linear transformation
from v into v.

Lemma: Let v be a vector space over the field F; let U, T and T be linear operators on v; let c be
1 2
an element of F.
(a) IU = UI = U;

(b) U(T + T ) = UT + UT ; (T + T ) U = T U + T U;
1 2 1 2 1 2 1 2
(c) C(UT ) = (eU) T = U(eT ).
1 1 1

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