Page 157 - DMTH502_LINEAR_ALGEBRA

Unit 11: The Double Dual

Notes
m
A g
ki j k
k 1
m
A
ki jk
k 1

A .
ji
For any linear functional f on V

m
f f i i . f ...(17)
i 1

t
t
If we apply this formula to the functional f T g j and use the fact that T g j i A ji , we have
n
t
T g A . f
j ji i ...(18)
i 1
from which it immediately follow that B A .
ij ji
t
Definition: If A is an m × n matrix over the field F, the transpose of A is n × m matrix A defined
by A t ij A ji .

Theorem 8: Thus states that if T is a linear transformation from V into W, the matrix of which in
some pair of bases is A, then the transpose transformation T is represented in the dual pair of
t
bases by the transpose matrix A . t
Theorem 9: Let A be any m × n matrix over the field F. Then the row rank of A is equal to the
column rank of A.

Proof: Let  be the standard ordered basis for F and ' the standard ordered basis for F . Let T
n
m

m
n
be the linear transformation from F into F such that the matrix of T relative to the pair  , ' is
A, i.e.,
. T x ,...,x y ,...,y
1 n 1 m
n
where y A x . ...(19)
i ij j
j 1
The column rank of A is the rank of transformation T, because the range of T consists of all
m-tuples which are linear combinations of the column vectors of A.

Relative to the dual bases '* and *, the transpose mapping T t is represented by the matrix A . t
Since the columns of A are the rows of A, we see by the same reasoning that the row rank of A
t
t
(the column rank of A ) is equal to the rank of T . By Theorem 7, T and T have the same rank, and
t
t
hence the row rank of A is equal to the column rank of A.

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