Page 336 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes In a similar manner, f determines a linear transformation R from V into V*. For each fixed in
f
V, f( , ) is linear as a function of . We define R ( ) to be the linear functional on V whose value
f
on the vector is f( , ).
Theorem 2: Let f be a bilinear form on the finite-dimensional vector space V. Let L and R be a
f f
linear transformation from V into V* defined by (L )( ) = f( , ) = (R )( ). Then rank (L ) = rank
f f f
(R ).
f
Proof: One can give a ‘coordinate free’ proof of this theorem. Such a proof is similar to the proof
that the row-rank of a matrix is equal to its column-rank. Some here we shall give a proof which
proceeds by choosing a coordinate system (basis) and then using the ‘row-rank equals column-
rank’ theorem.
To prove rank (L ) = rank (R ), it will suffice to prove that L and R have the same nullity. Let be
f f f f
an ordered basis for V, and let A = [f] . If and are vectors in V, with coordinate matrices X and
t
Y in the ordered basis , then f( , ) = X AY. Now R ( ) = 0 means that f( , ) = 0 for every in
f
t
V, i.e., that X AY = 0 for every n 1 matrix X. The latter condition simply says that AY = 0. The
nullity of R is therefore equal to the dimension of the space of solutions of AY = 0.
f
t
Similarly, L ( ) = 0 if and only if X AY = 0 for every n 1 matrix Y. Thus is in the null space of
f
t
t
L if and only if X A = 0, i.e. A X = 0. The nullity of L is therefore equal to the dimension of the
f f
space of solutions of A X = 0. Since the matrices A and A have the same column-rank, we see that
t
t
nullity (L ) = nullity (R ).
f f
Definition: If f is a bilinear form on the finite-dimensional space V, the rank of f is the integer
r = rank (L ) = rank (R ).
f t
Corollary 1: The rank of a bilinear form is equal to the rank of matrix of the form in any ordered
basis.
Corollary 2: If f is a bilinear form on the n-dimensional vector space V, the following are
equivalent:
(a) rank (f) = n
(b) For each non-zero in V, there is in V such that f( , ) 0.
(c) For each non-zero in V, there is an in V such that f( , ) 0.

Proof: Statement (b) simply says that the null space of L is the zero subspace. Statement (c) says
f
that the null space of R is the zero subspace. The linear transformations L and R have nullity 0
f f f
if and only if they have rank n, i.e., if and only if rank (f) = n.
Definition: A bilinear form f on a vector space V is called non-degenerate (or non-singular) if it
satisfies conditions (b) and (c) of Corollary 2.

If V is finite-dimensional, then f is non-degenerate provided f satisfies any one of the three
conditions of Corollary 2. In particular, f is non-degenerate (non-singular) if and only if its
matrix in some (every) ordered basis for V is a non-singular matrix.

Example 4: Let V = R , and let f be the bilinear form defined on = (x , ..., x ) and =
n
1 n
(y ..., y ) by
1 n
f( , ) = x y + ... + x y .
1 1 n n
n
Then f is a non-degenerate bilinear form on R . The matrix of f in the standard basis is the n n
identity matrix.
f(x, y) = X Y.
t

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