Page 145 - DMTH502_LINEAR_ALGEBRA

Unit 10: Linear Functionals

Now, Notes
n
f = ( f i )f i
i 1
so that if f is in W° we have (f i ) 0 for i k and

n
f
f = ( f i ) . i
i k 1
We have shown that if dim W = k and dim V = n then dim W° = n – k.
Corollary: If W is a k-dimensional subspace of an n-dimensional vector space V, then W is the
intersection of (n – k) hyperspaces in V.
Proof: This is a corollary of the proof of Theorem 3 rather than its statement. In the notation of
the proof, W is exactly the set of vectors such that ( ) 0,f i i k 1,..., . In case k = n – 1, W is the
n
null space of f .
n
Corollary: If W and W are subspaces of a finite-dimensional vector space, then W = W if and
1 2 1 2
0
only if W 0 W .
1 2
Proof: If W = W , then of course W 0 W 0 . If W 1 W 2 , then one of then two subspaces contains
1 2 1 2
a vector which is not in the other. Suppose there is a vector which is in W but not in W . By the
2 1
previous corollaries (or the proof of Theorem 3) there is a linear functional f such that ( ) 0f
0
0
for all in W, but ( )f 0. Then f is in W but not in W and W 0 W 0 .
1 2 1 2
10.2 System of Linear Equations
The first corollary says that, if we select some ordered basis for the space, each k-dimensional
subspace can be described by specifying (n – k) homogeneous linear conditions on the coordinates
relative to that basis.
Let us look briefly at system of homogeneous linear equations from the point of view of linear
functionals. Suppose we have a system of linear equations,

A x . . . A x 0
11 1
1n n
 
A x . . . A x 0
m 1 1 mn n
,
m
for which we wish to find the solutions. If we let f i 1,..., , be the linear functional on F n
i
defined by
x
f i ( ,....,x n ) A x ... A x
ix i
i
in n
then we are seeking the subspace of F of all such that
n
m
f i ( ) 0, i 1,..., .
In other words, we are seeking the subspace annihilated by f 1 ,..., f m . Row-reduction of the
coefficient matrix provides us with a systematic method of finding this subspace. The n-tuple
(A ,....A ) gives the coordinates of the linear functional f relatives to the basis which is dual to
1 i in i
the standard basis for F . The row space of the coefficient matrix may thus be regarded as the
n
space of linear functionals spanned by f 1 ,..., f m . The solution space is the subspace annihilated
by this space of functionals.

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