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
R
n
{\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.
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...
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) +...
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...