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Kunze Linear Algebra
· Problem 10.3.9
Bilinear Forms
Mathematics — Methods · skew-symmetric-product-functionals
Problem
Let
V
V
V
be a finite-dimensional vector space and
L
1
L_1
L
1
,
L
2
L_2
L
2
linear functionals on
V
V
V
. Show that the equation
f
(
α
,
β
)
=
L
1
(
α
)
L
2
(
β
)
−
L
1
(
β
)
L
2
(
α
)
f(\alpha, \beta) = L_1(\alpha)L_2(\beta) - L_1(\beta)L_2(\alpha)
f
(
α
,
β
)
=
L
1
(
α
)
L
2
(
β
)
−
L
1
(
β
)
L
2
(
α
)
defines a skew-symmetric bilinear form on
V
V
V
. Show that
f
=
0
f = 0
f
=
0
if and only if
L
1
L_1
L
1
,
L
2
L_2
L
2
are linearly dependent.
Related theory
Symmetric And Skew Symmetric Matrices
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