Page 346 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 31.3 Keywords

Skew Symmetric Bilinear Form: A bilinear form f on V is called skew symmetric if f( , ) –f( , )
for all vectors, , in V.

Skew-symmetric matrix: A matrix A in some (or every) ordered basis is skew-symmetric, if
+
A = –A, i.e. the two by two matrix
0 1
1 0

is a skew-symmetric matrix.
A non-degenerate skew-symmetric bilinear form f is such that

0, i j
f( , ) =
i j
1, i j
f( , ) = f( , ) = 0
i i i i
the dimension of the space must be even i.e. n = 2k.

31.4 Review Questions

1. Let V be a vector space over a field F. Show that the set of all skew-symmetric bilinear
forms on V a sub-space of L(V, V, F)
2. Let V be a finite dimensional vector space and L , L linear functional on V. Show that the
1 2
equation
f( , ) = L ( ) L ( ) – L ( ) L ( )
1 2 1 2
denotes a skew symmetric bilinear form on V. Also show that f = 0 if and only if L , L are
1 2
linearly dependent.

31.5 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael Artin, Algebra

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