Page 158 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Now we see that if A is an m ×n matrix over F and T is the linear transformation from F into F m
n
defined above, then
rank (T) = row rank (A) = column rank (A) ...(20)
and we shall call this number simply the rank of A.

Example 1: This example will be of a general nature – more discussion than example. Let
V be an n-dimensional vector space over the field F, and let T be a linear operator on V. Suppose
 = 1 ,..., n is an ordered basis for V. The matrix of T in the ordered basis  is defined to be the

n × n matrix A such that
n
T j A ij i ...(21)
j 1

in other words, A is the ith coordinate of the vector T j in the ordered basis . If f 1 ,..., f is
ij
n
the dual basis of , this can be stated simply

A ij f T j ...(22)
i
Let us see what happens when we change basis. Suppose

 '= ' 1 ,..., ' n

is another ordered basis for V, with dual basis f ' ,...,f ' . If B is the matrix of T in the ordered
1 n
basis ', then

'
B f T ' . ...(23)
ij i j
Let U be the invertible linear operator such that U j ' j . Then the transpose of U is given by

t
t –1
–1 t
U f t ' i . f It is easy to verify that since U is invertible, so is U and (U ) = (U ) . Thus
t
t
n
f ' U –1 , f i 1,..., . Therefore,
t i
t '
1
B U f T
ij i j
1
f U T '
i j
1
f U TU . ...(24)
i j
1
f U TU 1
Now what does this say? Well, i j is the i, j entry of the matrix of U TU in the ordered
basis  . Our computation above shows that this scalar is also the i, j entry of the matrix of T in
the ordered basis '. In other words

152 LOVELY PROFESSIONAL UNIVERSITY