Bilinear form Definition and 11 Threads

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 ##...

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...

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}##...

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...

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...

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...

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...

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...

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?

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...

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...