Page 332 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Let us formally define the Bilinear form as follows:
A Bilinear Form: Let V be a vector space over the field F, a bilinear form is a function f, which
assigns to each ordered pair of vectors , in V a scalar f( , ) in F, and which satisfies

( f c 1 2 , ) cf ( 1 , ) ( f 2 , )
...(1)
( f 1 c 1 , 2 ) cf ( 1 , 1 ) ( f 2 , 2 )
Thus a bilinear form on V is a function f from V V into F which is linear as a function of either
of its arguments when the other is fixed. The zero function from V V into F is clearly a bilinear
form. Also any linear combination of bilinear forms on V is again a bilinear form is f and g are
bilinear on V, so is cf + g where c is a scalar in F. So we may conclude that the set of all bilinear
forms on V is a subspace of the space of all functions from V V into F. Let us denote the space
of bilinear forms on V by L(V, V, F).

Example 1: Let m, n be positive integers and F a field. Let V be the vector space of all
m n matrices over F. Let A be a fixed m m matrix over F. Define
f (X, Y) = tr (X*AY)
A
then f is a bilinear form on V. For, if x, y, z are m n matrices over F,
A
+
f (CX, Z, Y) = tr [(CX + Z) AY]
A
t
t
= tr [cX AY] + tr [Z AY]
= cf (X, Y) + f (Z, Y)
A A
If we take n = 1, we have
f (X, Y) = X AY + A x y
t
ij i j
A
i j
So every bilinear form f for some A is of this form on a space of m × 1.
A
2
Example 2: Let F be a field. Let us find all bilinear forms on the space F . Suppose f is such
2
a bilinear form. If = (x , x ) and = (y , y ) are vectors in F , then
1 2 1 2
f( , ) = f(x + x , )
1 1 2 2
= x f( , ) + x f( , )
1 1 2 2
= x f( , y + y ) + x f( , y + y )
1 1 1 1 2 2 2 2 1 1 2 2
= x y f( , ) + x y f( , ) + x y f( , ) + x y f( , ).
1 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2
Thus f is completely determined by the four scalars A = f( , ) by
ij i j

f( , ) = A x y + A x y + A x y + A x y
11 1 1 12 1 2 21 2 1 22 2 2
= A x y
ij i j
, i j
If X and Y are the coordinate matrices of and , and if A is the 2 2 matrix with entries A(i, j) =
A = f( , ), then
ij i j
t
f( , ) = X AY. … (2)
We observed in Example 1 that if A is any 2 × 2 matrix over F, then (2) defines a bilinear form on
2
2
F . We see that the bilinear forms on F are precisely those obtained from a 2 × 2 matrix as in (2).

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