What is Bilinear form: Definition and 12 Discussions

In mathematics, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:

B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)The dot product on



{\displaystyle \mathbb {R} ^{n}}
is an example of a bilinear form.The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.
When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.

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  1. G

    Reduced equation of quadratic forms

    Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0 ##...
  2. Euler2718

    I Clarifying a corollary about Quadratic Forms

    The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...
  3. nightingale123

    I Why is there a Matrix A that satisfies F(x,y)=<Ax,y>?

    I'm having trouble understanding a step in a proof about bilinear forms Let ## \mathbb{F}:\,\mathbb{R}^{n}\times\mathbb{R}^{n}\to \mathbb{R}## be a bilinear functional. ##x,y\in\mathbb{R}^{n}## ##x=\sum\limits^{n}_{i=0}\,x_{i}e_{i}## ##y=\sum\limits^{n}_{j=0}\;y_{j}e_{j}##...
  4. caffeinemachine

    MHB Bilinear Form Non-Degenerate on a Subspace.

    I am trying to prove the following standard result:Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace of $V$ such that $f$ is non-degenerate on $W$. Then $$V=W\oplus W^\perp$$(Here $W^\perp=\{v\in...
  5. TheFerruccio

    Integrating until symmetric bilinear form

    Homework Statement I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here. Suppose I have a square domain over which I am integrating two functions w and...
  6. M

    How to visualise bilinear form and inner products?

    Hi I'm taking abstract linear algebra course and having trouble visualising bilinear form and inner products. I can visualise vector spaces, span, dimensions etc but haven't managed to figure out how to visualise this yet. Could someone please explain it to me in a visual way? I can't understand...
  7. S

    Which subspaces retain nondegeneracy of a bilinear form?

    Suppose I have a nondegenerate alternating bilinear form <,> on a vector space V. Under what conditions would a subspace U of V retain nondegeneracy? That is, if u ∈ U and u ≠ 0, then could I find a w ∈ U such that <u,w> ≠ 0? So for example, it's clear that no one-dimensional subspace W of V...
  8. O

    How do we get this equality about bilinear form

    Homework Statement B(u,u)=\int_{0}^{L}a\frac{du}{dx}\frac{du}{dx}dx B(.,.) is bilinear and symmetric, δ is variational operator. In the following expression, where does \frac{1}{2} come from? As i know variational operator is commutative why do not we just pull δ to the left? B(\delta...
  9. C

    Show that a bilinear form is an inner product

    Hi, I have a bilinear form defined as g : ℝnxℝn->ℝ by g(v,w) = v1w1 + v2w2 + ... + vn-1wn-1 - vnwn I have to show that g is an inner product, so I checked that is bilinear and symmetric, but how to show that it's nondegenerate too?
  10. S

    Bilinear Form & Linear Functional: Symmetric & Coercive?

    Homework Statement The bilinear form are symmetric, i.e. a(u,v) = a(v,u) for all u and v. Find the bilinear form and the linear functional for the problem -\Deltau + b . \nablau + cu = f(x) in \Omega u = 0 on the boundary. Is this bilinear form for this problem symmteric? Is it coersive...
  11. D

    If A(x,y) is a Positive Definate Bilinear Form then sqrt(A(x,x)) defines a norm

    Homework Statement The Problem is from Mendelson Topology. Let V be a vector field with the real numbers as scalars. He defines a bilinear form as a function A:V x V -> R s.t for all x,y,z an element of V and real numbers a,b,c A(ax +by, z) = aA(x,z) + bA(y,z) and A(x,by + cz) = bA(x,y) +...
  12. D

    What is the Matrix of a Non-Degenerate Non-Symmetric Bilinear Form?

    Hello I was reading through some research and I came across the proof of a lemma which I did not wholly understand. The problem statement is as follows: Let F be a non-degenerate non-symmetic bilinear form in V. Then there exists a basis in V with respect to which F has one of the following...