Let \(f\) be any bilinear form on a finite-dimensional vector space \(V\). Let \(W\) be the subspace of all \(\beta\) such that \(f(\alpha, \beta) = 0\) for every \(\alpha\). Show that \[ \operatorname{rank} f = \dim V - \dim W. \] Use this result and the result of Exercise 9 to compute the rank of the bilinear form defined in Exercise 6.
Solution
1. Construct the associated linear map into the dual space \(V^*\):
1.1. For each \(\beta \in V\), define \(L_\beta: V \to F\) by \(L_\beta(\alpha) = f(\alpha, \beta)\).
linearity of \(f\) in the first slot implies \(L_\beta \in V^*\).
1.2. Define the map \(T: V \to V^*\) by \(T(\beta) = L_\beta\).
definition of \(T\).
1.3. \(T\) is linear: for all \(\beta_1, \beta_2 \in V\), \(c \in F\), and \(\alpha \in V\):\[ T(c\beta_1 + \beta_2)(\alpha) = f(\alpha, c\beta_1 + \beta_2) = c f(\alpha, \beta_1) + f(\alpha, \beta_2) = c T(\beta_1)(\alpha) + T(\beta_2)(\alpha). \]
linearity of \(f\) in the second slot.
2. Identify the kernel and rank of \(T\):
2.1. Compute the kernel of \(T\):\[ \ker(T) = \{\beta \in V : T(\beta) = 0\} = \{\beta \in V : f(\alpha, \beta) = 0 \text{ for all } \alpha \in V\} = W. \]Thus \(\operatorname{nullity}(T) = \dim W\).
definition of \(W\).
2.2. Let \(\mathcal{B} = \{v_1, \dots, v_n\}\) be an ordered basis of \(V\), with dual basis \(\mathcal{B}^* = \{v_1^*, \dots, v_n^*\}\).The matrix of the bilinear form \(f\) relative to \(\mathcal{B}\) is \(A = [a_{ij}]\) with \(a_{ij} = f(v_i, v_j)\).
definition of matrix of a bilinear form.
2.3. For each basis vector \(v_j\), \(T(v_j) = \sum_{i=1}^n f(v_i, v_j) v_i^* = \sum_{i=1}^n a_{ij} v_i^*\).Thus the coordinate vector \([T(v_j)]_{\mathcal{B}^*}\) is the \(j\)-th column of \(A\).
coordinate evaluation on dual basis.
2.4. Therefore \(\dim(\operatorname{im} T) = \operatorname{rank}(A) = \operatorname{rank}(f)\).
rank of a linear map equals the column rank of its matrix.
3. Apply the Rank-Nullity Theorem:
3.1. By the Rank-Nullity Theorem applied to \(T: V \to V^*\):\[ \dim V = \operatorname{rank}(T) + \operatorname{nullity}(T) = \operatorname{rank}(f) + \dim W. \]
Rank-Nullity Theorem for linear transformations on finite-dimensional spaces.
3.2. Rearranging gives:\[ \operatorname{rank}(f) = \dim V - \dim W. \]
algebraic rearrangement.
4. Conclusion:\(\operatorname{rank}(f) = \dim V - \dim W\). Q.E.D.
1 through 3.