๐ What is a Bilinear Form?
A bilinear form is essentially a function that takes two vectors as input and spits out a scalar. The crucial part? It's linear in both arguments. Think of it as a way to measure how two vectors interact.
- ๐ Definition: A bilinear form on a vector space $V$ over a field $F$ is a function $B: V \times V \rightarrow F$ such that for all vectors $u, v, w \in V$ and scalars $a, b \in F$:
- โ $B(au + bv, w) = aB(u, w) + bB(v, w)$ (Linearity in the first argument)
- โ $B(u, aw + bv) = aB(u, w) + bB(u, v)$ (Linearity in the second argument)
- ๐งฎ Example: The dot product in $\mathbb{R}^n$ is a classic example of a bilinear form.
๐ What is a Quadratic Form?
A quadratic form, on the other hand, takes a single vector as input and produces a scalar. Itโs derived from a bilinear form by evaluating that bilinear form with the same vector in both input slots. In other words, itโs a function that gives you a value related to the โsizeโ or โmagnitudeโ of a vector, possibly with respect to some direction.
- ๐ Definition: A quadratic form on a vector space $V$ over a field $F$ is a function $Q: V \rightarrow F$ such that $Q(v) = B(v, v)$ for some bilinear form $B: V \times V \rightarrow F$.
- ๐ก Example: $Q(x, y) = x^2 + 2xy + y^2$ is a quadratic form derived from the bilinear form $B((x_1, y_1), (x_2, y_2)) = x_1x_2 + x_1y_2 + y_1x_2 + y_1y_2$.
๐ Quadratic Form vs. Bilinear Form: Key Differences
Let's highlight the core distinctions using a table:
| Feature |
Bilinear Form |
Quadratic Form |
| Input |
Two vectors ($V \times V$) |
One vector ($V$) |
| Output |
A scalar ($F$) |
A scalar ($F$) |
| Linearity |
Linear in both arguments |
Not linear |
| Relationship |
Fundamental; can be used to define a quadratic form |
Derived from a bilinear form |
| Representation |
Can be represented by a matrix |
Can be represented by a matrix (symmetric) |
| Symmetry |
Can be symmetric, skew-symmetric, or neither |
Implies an underlying symmetric bilinear form |
๐ Key Takeaways
- ๐ฏ Bilinear forms take two vectors and produce a scalar, showing linearity in both.
- ๐ Quadratic forms take one vector and produce a scalar, derived from a bilinear form evaluated with the same vector twice.
- โ Polarization Identity: A symmetric bilinear form can be recovered from its associated quadratic form using the polarization identity: $B(u, v) = \frac{1}{2}[Q(u+v) - Q(u) - Q(v)]$.
- ๐ Applications: Bilinear forms are used in defining inner product spaces and tensors. Quadratic forms are used in optimization problems and conic sections.