a bilinear form is always the sum of two others

Question

Let $K$ be a field with a characteristic, other than 2. Let $V$ be a finite dimensional vector space over $K$, and let $\gamma: V \times V \to K$ be a bilinear form.

I now want to show that there exist unique bilinear forms $\gamma_s, \gamma_a: V \times V \to K$ that satisfy the following properties:

1) $\gamma_s$ is symmetric.

2) $\gamma_a$ is antisymmetric.

3) $\gamma = \gamma_s + \gamma_a$.

Thanks in advance. I'm not very used to working with bilinear forms.

Answer 1

Every bilinear form can be written as a sum of a symmetric and antisymmetric form in a unique way:

$$B=\frac{B+B^T}{2}+\frac{B-B^T}{2}.$$