Bilinear form

This is a specialisation of a Bilinear map and a generalisation of Inner product

Definition

Let $(V,F)$ be a vector space over a field $F$, a mapping:

• $$\langle\cdot,\cdot\rangle:V\times V\rightarrow F$$

is a bilinear form if:

• It is bilinear, that is to say:
  • It is linear in each coordinate, which is to say:
    • $$\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle\alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle]$$ and
    • $$\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle]$$

Properties

We say the bilinear form is property when it has any of the following properties:

Property	Definition	Comment
Symmetric	$$\forall x,y\in V[\langle x,y\rangle=\langle y,x\rangle]$$
Skew-symmetric or Antisymmetric	$$\forall x,y\in V[\langle x,y\rangle=-\langle y,x\rangle]$$	I use antisymmetric
Alternate or Alternating	$$\forall x\in V[\langle x,x\rangle=0]$$	I use alternating. $0$ denotes the additive identity of $F$

See also

• Inner product
• Bilinear map
• Linear map

---

TODO: Unite with inner product over real or complex field. Page 260 in Roman and page 206 are what is needed

---