Find a basis for the space of all skew-symmetric bilinear forms on \(\RR^n\).
Solution
1. A skew-symmetric bilinear form \(f\) on \(\RR^n\) is determined by its matrix \(A = (a_{ij})\) with \(a_{ij} = f(e_i, e_j)\), where \(A\) is skew-symmetric (\(A^t = -A\)).
bilinear forms correspond to matrices, and skew-symmetry means \(a_{ij} = -a_{ji}\).
2. Hence the space of skew-symmetric forms is isomorphic to the space of skew-symmetric \(n \times n\) matrices, of dimension \(\binom{n}{2}\).
a skew-symmetric matrix is determined by its strictly upper-triangular entries \(a_{ij}\) for \(i < j\), of which there are \(\binom{n}{2}\).
3. For each pair \(1 \le i < j \le n\), define the form \(f_{ij}\) by\[f_{ij}(x, y) = x_i y_j - x_j y_i.\]
definition.
4. The matrix of \(f_{ij}\) is \(E_{ij} - E_{ji}\) (where \(E_{ij}\) is the matrix with a \(1\) in position \((i,j)\) and \(0\) elsewhere).
\(f_{ij}(e_p, e_q) = \delta_{pi}\delta_{qj} - \delta_{pj}\delta_{qi}\).
5. The set \(\{f_{ij} : 1 \le i < j \le n\}\) is a basis.
the matrices \(E_{ij} - E_{ji}\) (for \(i < j\)) form a basis of the skew-symmetric matrices, and 2 gives the dimension \(\binom{n}{2}\), matching the number of these forms.
6. Q.E.D.
5.