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Questions tagged [bilinear-form]

A bilinear form over an $F$-vector space $V$ is a mapping $B:V\times V\to F$ that is linear in each of its arguments, when the other argument is held fixed.

A bilinear form on a vector space $V$ over a field $F$ is a mapping $B:V\times V \to F$ that is separately linear in each of its arguments.

$B$ is called symmetric if $B(v,w) = B(w,v)$ for all $v,w\in V$. $B$ is called alternating if all vectors are isotropic, i.e. if $B(v,v) = 0$ for all $v\in V$.

1162 questions
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Orthogonal Bases w.r.t a given Bilinear Form.

Under what conditions there exists an orthogonal basis? Or even better, is there a characterization of the existence of an orthogonal basis in terms of a given bilinear form and/or the base field? For instance, if the characteristic of the field is…
user 1987
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Skew-symmetric bilinear form

Let $V$ be a vector space over K and $B: V × V \to K$ a bilinear form on $V$. $\beta$ is a basis of $V$ and $G$ the gram matrix of $B$ with respect to $\beta$. How to proof: $B$ is skew-symmetric $\Leftrightarrow$ $G=-G^T$? For $\Rightarrow$ I…
Humertun
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If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that

If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that $$\lim\limits_{(h,k) \to (0,0)} \dfrac{|f(h,k)|}{|(h,k)|} = 0$$.
Ri-Li
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Question about bilinear form

Prove that every bilinear form $f:\mathbb R^n \times \mathbb R^n\rightarrow \mathbb R$ has a basis $\{v_1,\ldots,v_n\} \subset \mathbb R^n$ such that $f(v_i,v_j)=-f(v_j,v_i)$ for every $i\neq j$. I have no idea how to start thinking about this one.…
Kob1
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Bilinear form and weak sector condition

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a bilinear map. For $\alpha\geq0$ we…
ko4
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Coercive vs positive definite for a symmetric bilinear form

I was wondering once again what the difference in concept of positive definite and coercive was. Of course, if you have a bilinear form $a(.,.)$, that is coercive, positive definiteness follows directly from the…
martin
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Bilinear form Find q $\in\ V$ such that $\phi(p)=\psi(p,q)$ for all $p\in V$

Let $V$ be the $\mathbb{R}$-Vector space of polynomials with degree $\leq$ 2. Let $\psi\colon V \times V \rightarrow \mathbb{R}$ be the bilinear form $$(p,q) \mapsto \int_{0}^{1}p(x)q(x)dx$$ Let $\phi$ be a linear form $p \mapsto p(0)$. Find a $q…
Lunnnniii17289
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Simplifying equation

I'm looking at answers to a question and I see these two lines of the equation. How is they're getting the -0.5095 on the bottom? eqn
controlled_
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Prove that for every $f$, a bilinear form, there exists a basis ${v_1,...,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$

I didn't want to bloat the title, i'll also add that the basis is in ${\mathbb{R}^n}$. This question seems kind of easy, but its from a test so I assume there is a catch here somewhere. I tried to prove it by contradiction. Assume by contradiction…
Xsy
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Scalar / Dot product

I have a simple question about Scalar / Dot product. (http://en.wikipedia.org/wiki/Dot_product) Say f is a bilinear form. I have to tell if f defines a dot product. I didn't understand what I should do, what does f has to satisfy so it can be called…
Jenni201
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Application of a bilinear form on a function on $S^{n-1}$

Suppose we have a non singular bilinear application $\mu$:$\mathbb{R}^n \times \mathbb{R}^n \rightarrow\mathbb{R}^n$ with $\mu(v,w)=\mu (w,v)$. Let the continous application $g: S^{n-1} \rightarrow S^{n-1}$ given by $g(x)= \frac{\mu(x,x)}{||…
usere5225321
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Bilinear form matrix

So I read on one of the answers here that β(x,y)=xTAy where A is a matrix. So if I'm reading this right then the matrix of the bilinear form corresponding to the dot product would be the identity Matrix?
R. Emery
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Citation for rewriting non-degenerate bilinear form so that the Gram matrix is an anti-diagonal

For a paper, I am looking for a reference in the literature for the following theorem: Given a symmetric non degenerate bilinear form $\phi$ on a finite dimensional vector space , with $\dim >1$, and $\phi(e_i, e_j)=0$ if $i+j>n+1$, there is a…
number.seven
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bilinear symmetric form

Let $b$ be a symmetric bilinear form on a $k$-dimensional vector space $V_k(\mathbb{F}_q)$, with $char(\mathbb{F}_q)=2$ and $k$ odd. Theorem 3.15 of the book S. Ball, Finite Geometry and Combinatorial applications states that $$ V_k(\mathbb{F}_q)=E…
Mathsa
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Are there polarization-like identities for nonsymmetric bilinear forms?

Let $B$ be a bilinear form on a vector space $V$. If $B$ is symmetric, there are polarization identities, e.g. $$2B(u,v)=B(u+v,u+v)-B(u,u)-B(v,v).$$ Are there any identities, similar in character, for $nonsymmetric$ ($B(u,v)\neq B(v,u)$ for some…
Stuck
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