Page 134 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes and so it must be that
T '  P 1 T  . P ...(8)

This answers our questions.
Before stating this result formally, let us observe the following. There is a unique linear operator

U which carries  onto ', defined by
U ' , j = 1,...,n
j j
This operator U is invertible since it carries a basis for V onto a basis for V. The matrix P (above)
is precisely the matrix of the operator U in the ordered basis . For, P is defined by

n
' P
j ij i
i 1

and since U ' , this equation can be written as
j j
n
U j P ij i .
i 1

So P = U , by definition.

Theorem 4: Let V be a finite-dimensional vector space over the field F, and let



 B = =  1 ,..., n and ' = ' 1 ,..., ' n
be ordered basis for V. Suppose T is linear operator on V. If P = P 1 ,...,P n is the n×n matrix with
-1
columns P P T , then
j 
T '  P 1 T  . P

n
Alternatively, if U is the invertible operator on V defined by U ' , j 1,..., then
j j
T U 1 T U .
'  '   

Self Assessment

1. Let T be the linear transformation T R 3 R 3 , defined by
:
T (x,y,z) = (2y+z, x–4y, 3z)
find the matrix T, with respect to the basis

E 1,1,1 ,E 1,1,0 and E 1,0,0
1 2 3
2. A transformation T is defined by

1
T , x y = x –y x y
,
2

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