symmetric bilinear form
Imagine measuring how two vectors ‘relate’ in a way that does not care which one you name first. The dot product does exactly this: u · v equals v · u. A symmetric bilinear form abstracts this even-handedness — swapping the two inputs never changes the answer. This is the kind of pairing that supports notions of length, angle, and orthogonality.
Formally, a bilinear form B on V is symmetric if B(u, v) = B(v, u) for all u, v in V. In terms of any Gram matrix G this says G = G^T: the matrix is symmetric. Symmetric bilinear forms are exactly the forms that arise by polarizing a quadratic form (when the characteristic is not 2), so the theory of symmetric forms and the theory of quadratic forms are two views of one subject.
Two vectors are called orthogonal when B(u, v) = 0. Orthogonality lets one build orthogonal bases and split V into pieces, leading to diagonalization, the rank and signature, and ultimately a classification of forms. In characteristic 2 the link to quadratic forms breaks down — symmetric and alternating coincide on the diagonal — so quadratic forms must then be treated as primary data in their own right.
B(x, y) = x_1 y_1 + x_2 y_2 - x_3 y_3 on R^3 has Gram matrix [1,0,0; 0,1,0; 0,0,-1], which equals its transpose. Here e_1 and e_3 are orthogonal, and e_3 is orthogonal to itself only if its self-value were 0 — it is -1, so e_3 is not isotropic.
A diagonal symmetric form of signature (2, 1).
Symmetric and alternating bilinear forms are complementary species: in characteristic not 2, every bilinear form is uniquely a symmetric part plus an alternating part, where alternating means B(v, v) = 0 for all v.