I am trying to understand what is meant by a bilinear form. It seems to be some constraint on a map of two vectors to a field. I would be grateful if someone could give me a very trivial specific (as in non-abstract) example of what is, and isn't a bilinear form. I find examples more insightful than definitions.
What use are bilinear forms?
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$\begingroup$A bilinear map is a special case of multilinear maps
To say that $f$ is bilinear means that:
- $f(\alpha x+\beta y,z)=\alpha f(x,y)+\beta f(y,z)$ and
- $f(a,\alpha b+\beta c)=\alpha f(a,b)+\beta f(a,c)$
For $x,y,a\in V$ and $z,b,c\in U$ where $U,V$ are vector spaces over a field, $F$ and $\alpha,\beta\in F$
You can find examples online, but here are a few.
First:
Any inner product is a bilinear form (this includes the familiar dot-product)
Second
$\endgroup$ 7 $\begingroup$Any $n\times n$ matrix is a bilinear form if you consider it as: $M(x,y)=x^TMy$
- An inner product over the real numbers is a positive-definite symmetric bilinear form. Since we define $\langle \cdot, \cdot\rangle : V \times V \to \mathbb R$ and it holds the properties
$$\langle u + v, w\rangle = \langle u,w\rangle + \langle v,w \rangle$$
and $$\langle \alpha u , v\rangle = \alpha \langle u,v\rangle$$
for any $u,v,w \in V$ and $\alpha \in \mathbb R$.
- Given real-valued functions $f: E \to \mathbb R$ and $g : F \to \mathbb R$ and the function $b : E \times F \to \mathbb R$ defined as $b (u,v) = f(u)g(v)$ is also a bilinear form known as tensor product. Particularly, if $E = F$ then for $f,g: E \to \mathbb R$
$$(f \cdot g) (u,v) = f(u)g(v) + f(v)g(u)$$ and
$$(f \wedge g)(u,v) = f(u)g(v) - f(v)g(u)$$
are bilinear forms, the first symmetric, the second anti-symmetric.
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