# Tagged Questions

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.

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### An example of symmetric, coercive, discontinuous bilinear form over a Hilbert space?

Can you show me an example of symmetric, coercive and discontinuous bilinear form over a Hilbert space? I saw some stuff here Give an example of a discontinuous bilinear form. but the forms there are ...
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### Isometry on the spanning set is an isometry, intuition.

I was reading and came across the following statement. Let $C$ and $D$ be vector spaces equipped with bilinear forms, and $F: C \to D$ a linear map. Say $X_i$ is a spanning set of vectors for $C$ ...
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### Confusion over a change of bases

Suppose I am given that a bilinear form $\phi(u,v)$ on $\mathbb{R}^3$ is represented by a diagonalisable matrix $M$ with respect to the pair of bases $\{u_i\}$ and $\{v_i\}$. If I wanted to find a ...
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### Quadratic Form - find a minimal scalar $m \in \Bbb R$ such that $q(x,y,z) \le m(x^2+y^2+z^2)$

Let $q (x,y,z)$ be a quadratic form, $$q(x,y,z)=2zx+4yz-2xy$$ $$V=\Bbb R^3$$ Find a minimal scalar $m \in \Bbb R$ such that $$q(x,y,z) \le m(x^2+y^2+z^2)$$ for all $x,y,z \in \Bbb R$. ...
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### How is $f((x_1.y_1),(x_2,y_2)) = x_1y_1 + x_2y_2$ a bilinear form?

In section $10.1$ of Hoffman-Kunze's Linear Algebra, exercise $2$ states the following: Let f be the bilinear form on $\mathbb{R}^2$ defined by $f((x_1,y_1),(x_2,y_2)) = x_1y_1 + x_2y_2.$ Find the ...
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### Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W)$.

I have to prove the following question, but I'm having trouble. Any help would be greatly appreciated. Let $V$ and $W$ be two (not necessarily finite dimensional) vector spaces. Show there is a ...
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### Skew-Hermitian matrices

I have a couple of questions regarding skew-Hermitian matrices over finite fields. A matrix $A$ over $\mathbb{F}_{q^{2}}$ is skew-Hermitian if $A + A^{*} = 0$, where $A^{*}$ is the conjugate ...
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### Bijection between left integrals and right orthogonal associative bilinear forms in a bialgebra

Suppose $H$ is a bialgebra over a PID $R$. I am trying to understand the bijection between right orthogonal associative bilinear forms on $H^*$ and left integrals of $H$. Specifically, I have trouble ...
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### Bilinear forms (Existence of linear operator)

Can anybody help me in construction of linear operators in exercise 12 of section 10.1, Hoffman and Kunze. Let $f,g$ be bilinear forms on a finite dimensional vector space $V$. Suppose $g$ is non ...
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### How to determine the following statement is bilinear mapping or not?

I have already checked wikipedia about bilinear mapping but still I haven't figured out how these things are working. Could anyone explain me? Thanks in advance for your help. bilinear mapping or ...
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### Find the Rank and Signature of a Billinear Form

Let $V \in M_{2 \times 2} ^C$ be the set of all the herimitian of order 2. V is a linear space over $\Bbb R$. Check that $q(A) =2det(A)$ is a Square Billinear Form. In addition, Find ...
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### The trace of $F|K$ is non degenerate if and only if the trace of $E\otimes_{K}F|E$ is nondegenerate

Let $F|K$ be a finite field extension. I want to prove that if $E|K$ is an algebraic extension, then the bilinear form $$F\times F\rightarrow K, (x,y)\mapsto Tr_{F|K}(xy)$$ is nondegenerate if ...
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### Discontinuous bilinear form separately continuous

Do you have an example of a real normed space $V$ and a bilinear form $B : V \times V \to \mathbb R$ that is discontinuous but such that $B$ is separately continuous for each variable? $V$ has to be ...
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### Find a polarisation in a complex vector space.

I have trouble with the following problem Let $V$ be a complex vector space, of real dimension $2d$, and let $L$ be a lattice in $V$, that is the abelian group generated by a basis of $V$ as a real ...
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### Existence of linear operator

Let f and g be bilinear forms on FDVS V. Suppose g is non-degenerate. Show that there exists unique linear operator S and T on V such that $\ f(a,b)=g(Sa,b)=g(a,Tb)$ for all a,b What happens if g ...
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### Existence of linear functionals

Let $f$ be any skew-symmetric bilinear form on $\Bbb R^3$. Prove that there exist linear functionals $L$ and $M$ on $\Bbb R^3$ such that $$f(a,b)=L(a)M(b)-L(b)M(a)$$ I'm having a similar problem ...
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### Let f,g be bilinear forms on a finite dimensional vector space. Show that there exist unique linear operators

Let f,g be bilinear forms on a finite dimensional vector space. (a) Suppose g is non-degenerate. Show tha there exist unique linear operators T1, T2 on V such that f(a,b)=g(T1a,b)=g(a,T2b) for all ...
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### Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
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### Identity is the only real matrix which is orthogonal, symmetric and positive definite

All I could get using above information was that A^2=I, hence it is its own inverse. Using the fact that A is positive-definite I got that all diagonal entries will be greater than 0, but how does ...
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### Linear Functional from Bilinear form

Let $w$ be a bilinear form on $U\times V$ and $y_0$ belongs to $V$. If $y$ is a function defined on $U$ by $y(x)=w(x,y_0)$, then $y$ is a linear functional on $U$. Is it true that if $w$ is a ...
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### How to understand the geometry of bilinear forms that are not positive-definite?

I simply cannot find a good resource that explains intuitively how to understand the geometry that is induced on a vector space when the bilinear form is not positive-definite. In the ordinary ...
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### Positive definite quadratic form . Riemannian manifolds

Does anybody know how to solve it? I've done a lot of tries but I didn't succeeded Let $H^n=\{ (x_0,x_1,...,x_n)\in \Re^{n+1}:x_0^2+x_1^2+...+x_n^2=-1,x_0>0\}$ and the symetric bilinear form ...
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### Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
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### a symmetric bilinear form has a basis such that it's matrix with respect to it is diagonal

I'm reviewing a proof regarding $f$, a symmetric bilinear form having a basis such that it's matrix with respect to this basis is diagonal. Here's a summarization: For $n=1$ there's nothing to ...
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### Dual space isomorphism and the dual representation

Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate ...
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### Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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### Intersection form on $S^2 \tilde \times S^2$

The intersection form on the nontrivial $S^2$-bundle over $S^2$, denoted $S^2 \tilde \times S^2$, can be written as $\left[\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right]$ with ...
$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ... 0answers 37 views ### The space of alternating multilinear maps and existence of a bilinear map [duplicate] Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let$E$be a finite dimensional vector space over field$\mathbb R$with$E^*$as ... 1answer 42 views ### Example of where the sum of a subspace and its orthogonal complement is not the original vector space? Suppose$\mathbb{F}$is an arbitrary field and let$W$be a subspace of$\mathbb{F}^n$.$W^\perp$can be defined in exactly the same way as in the real case. Show by example that it isn't ... 0answers 66 views ### The Lax-Milgram Theorem for Banach Spaces I wish ask by Question. A significaive (this is for Banach and not Hilbert space, for example a$L_p$space with$p\neq 2$or another as you see, a real Banach space) and understable (constructive ... 1answer 21 views ### Find if a form is symmetric or skew-symmetric Consider the set of all n × n matrices in R. Given the defined function Φ:$M$(n,n)×$M$(n,n) → R , which Φ(A,B) =$tr$(A$^T$JB) , where J is a skew-symmetric n × n matrix , define if Φ is a ... 0answers 18 views ### Relation between bilinear symplectic forms and symplectic matrices 1. Symplectic Forms Let$F : \mathbb{K}^{2n} \times \mathbb{K}^{2n} \to \mathbb{K}$be a bilinear skew-symmetric nondegenerate form (as known as symplectic form). Then$F(u,v) = u^TAv$where$A = ...
I would like to describe the space of solutions to the following matrix equation. Here $U$ and $V$ are two unknown real matrices with $n$ lines and $p$ columns (i.e. they belong to ...