Bilinear form

A function of two vector inputs that is linear in each input when the other is fixed. It encodes geometric or algebraic structure through its values on pairs of vectors.

Linear map: A function between vector spaces that preserves vector addition and scalar multiplication. Fixing either input of a bilinear form produces a linear map.
Symmetric bilinear form: A bilinear form whose value is unchanged when its two inputs are exchanged. Symmetry makes the form correspond directly to a quadratic form over fields of characteristic not two.
Inner product: A positive-definite symmetric bilinear form that assigns lengths and angles to vectors. An inner product is a specialized bilinear form with positivity and symmetry.
Sesquilinear form: A function of two vector inputs that is linear in one input and conjugate-linear in the other. Over complex vector spaces, conjugation distinguishes sesquilinear forms from bilinear forms.
Dual space: The vector space of linear functionals from a vector space to its field of scalars. A bilinear form assigns a linear functional to each fixed vector in either input.
Alternating bilinear form: A bilinear form that evaluates to zero whenever both inputs are the same vector. Alternation forces skew-symmetry and underlies symplectic geometry.
Quadratic form: A homogeneous degree-two function on a vector space, commonly obtained by evaluating a symmetric bilinear form twice on one vector. A symmetric bilinear form determines a quadratic form through its diagonal values.
Hermitian form: A complex sesquilinear form whose value reverses to its complex conjugate when its inputs are exchanged. It replaces real symmetry with conjugate symmetry over complex vector spaces.
Tensor product: A vector space that represents bilinear maps as linear maps from pairs of vectors. Every bilinear form corresponds to a linear functional on the tensor product of its two input spaces.
Nondegenerate bilinear form: A bilinear form whose only vector pairing to zero with every vector is the zero vector. Nondegeneracy lets the form identify a vector space with its dual.

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