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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Find the linear maps that preserve this bilinear form.

This time I want to know which linear maps are the ones that preserve the following bilinear form: $$\beta(x,y)=2(x_1y_1-x_2y_2)$$ and they give as a hint the matrix: $$\begin{pmatrix} ...
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Query on Bilinear form

We are given a bilinear form Q(x,y) = x'Ay. Suppose Q(x,y)>=0 and A is non-singular. So can we then say that x'y>=0 ? Is there any result/lemma to prove this claim ?
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Let $f_1,f_2,f_3 \neq 0$ linear operators in $\mathbb{R}³$, then $\exists y\in\mathbb{R}³,y\neq0 $, {$f_1(y),f_2(y),f_3(y)$} is l.d.

What I did: Let $f_1,f_2,f_3: \mathbb{R}³ \rightarrow \mathbb{R}³$ linear operators and take $\exists y\in\mathbb{R}³,y\neq0 $. Let $\alpha, \beta, \gamma \in \mathbb{R}$ such that $\alpha f_1(y) + ...
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Bilinear transformation not working

I want to try this bilinear transformation of a rectangle to a quad described here http://www.fmwconcepts.com/imagemagick/bilinearwarp/FourCornerImageWarp2.pdf on page 4. I have the square ...
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How to find $f$ for a symmetric bilinear form?

Let's say we have the symmetric matrix:$$A = \left(\begin{array}{cc} 1&2 \\ 2&0 \end{array}\right)$$ How do I find the symmetric bilinear form of this $A$?
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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 ...
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Natural bilinear map $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$

$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ...
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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 ...
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35 views

Basis Bilinear map $g:U \times V \rightarrow W$ [closed]

I am having some trouble proving this: For a pair $(W,g:U \times V \rightarrow W)$ with $g$ bilinear, if $\{u_i\}$ is a basis for U and $\{v_j\}$ is a basis for $V$, then $\{g(u_i,v_j)\}$ is a ...
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35 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 ...
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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 ...
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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 ...
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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 = ...
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Describe the space of solutions to a simple matrix equation

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 ...
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Matrix representation of a bilinear form

Let $a:\,\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a bilinear operator (i.e. $a$ is a linear operator in each component). Then exists a square matrix $A\in \mathbb{R}^{n\times n}$ such ...
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Uniqueness of positive-definite subspace

Let $g$ be a bilinear form over $\mathbb{R}^8$ with signature $(5,3)$. True or false - There exists a unique five dimensional subspace such that $g$ is positive-definite on this subspace. My ...
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Linear independence of isotropic vectors relative to a bilinear form.

Let $V$ be a vector space over $\mathbb{F}$, and let $g: V\times V\to \mathbb{F}$ be a bilinear form. A vector $v\in V$ is isotropic if $g(v,v) = 0$. Let $v,w\in V$ be 2 isotropic vectors such that ...
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differentiability of a function consisting of a bilinear form

Let $A \in M_{n, n}(\mathbb{R})$ be a symmetric matrix, and let $N = \{x \in \mathbb{R}^n \mid x^t A x = 0\}$. I first want to show that $N$ is closed. Next, I want to find an explanation why $f(x) = ...
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Bilinear forms have same rank

Exercise 12, Chapter 10, sec 10.4 [Hoffman and Kunze]. Let $V$ be a finite-dimentional vector space over a subfield of the complex numbers, let f,g be skew-symmetric bilinear forms on V.Show that ...
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Non-degenerated restricted bilinear form

Studying bilinear forms, this question came to my mind: If $f$ is a non-degenerated bilinear form in $V$ and $W$ is a subspace of $V$, is $f$ restricted to $W$ is also non-degenerated? I know that ...
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Showing that a bilinear form is non degenerate

Given a finite dimensional vector space $V$ over $F$ and a fixed matrix $(\alpha_{ij})=A \in M_n(F)$ and the bilinear form on $V \times V$ by $B(u,v)=\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_{ij} ...
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Linear transformation and bilinear forms

I am trying to do the next exercise (Exercise 11, section 10.2, Hoffman & Kunze). I defined a linear functional $L: V \to V^*$ as $(L_f\alpha)(\beta)=f(\alpha, \beta)$. It is easy to show that ...
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Generalized criterion for positive definiteness of real symmetric matrices [duplicate]

I searched for the following question on stack exchange but couldnot find the general case answered or asked anywhere. This is an exercise from Artin's book, under the chapter named "Bilinear forms''. ...
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On bilinear forms: if $B$ is non-degenerate, then $(U_1\cap U_2)^{\perp_L} = U_1^{\perp_L} + U_2^{\perp_L}$

I think I have a proof, but I'm not completely certain it is correct: Because $B$ is non-degenerate, it follows that if $W \leq V$, then $W = W^{\perp_L \perp_R}$. It is evident, too, that $(U_1 + ...
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Complement of the radical of a bilinear form

The following is given(V is a vector space and $\gamma$ is a bilinear form: $rad(V)=\{v\in V|\;\gamma(v,w)=0 $ for all $ w \in W\}$ Let U be the complement of rad(V), show that ...
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determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
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Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
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Properties of bilinear forms

Let $V$ be a real vector space with norm $\|\cdot\|_V$ and $W$ a closed, linear subspace of $V$. A bilinear form $a\colon V\times V\rightarrow \mathbb{R}$ is called symmetric, if $a(u, v) = ...
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Show that a field extension $L/K$ is separable iff the trace form is non-degenerate.

Let $L$ be a finite field extension of $K$. I have the following question: Show that $L/K$ is separable if and only if the bilinear trace form $\text{Tr}_{L/K}:L\times L \to K$ is non-degenerate. ...
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isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
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Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
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A Theorem on Skew-Symmetric Bilinear Forms in Hoffman and Kunze's Book

On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads: Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew ...
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If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
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a bilinear form is always the sum of two others

Let $K$ be a field with a characteristic, other than 2. Let $V$ be a finite dimensional vector space over $K$, and let $\gamma: V \times V \to K$ be a bilinear form. I now want to show that there ...
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Surjectivity of an alternating, non degenerate bilineair form

I have a function $\varphi: V \times V \to k$ where $V$ is a vector space over ground field $k$ (although I would also like to prove this for $R$-modules). I know that $\varphi$ is alternating, that ...
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Find all complex matrices $A$ such that $n\operatorname{Tr}(AB) = \operatorname{Tr}(A)\operatorname{Tr}(B)$ for all $B$. [duplicate]

Consider a bilinear form $f(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A)\operatorname{Tr}(B)$ defined on $M_n(\mathbb{C})$. I need to find the set $U^\perp$ of all matrices $A$ such that ...
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bilinear forms on $M_{n, n}(K)$

Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set: $\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$. I now ...
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Show that a bilinear form is complex skew-Hermitian.

A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form $s : V × V → \mathbb C$ such that $$s(w,z) = -\overline{s(z, w)}.$$ Prove that the ...
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Matrix Representation of Bilinear Form/Map in Matrix Space

I am trying to understand bilinear forms and have a related problem, but unfortunately all of the examples I've been able to find deal with $V = \mathbb{R}^n$. I am dealing with $V = M_{n \times ...
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Necessary and sufficient condition for a bilinear form to be symmetric

Given the bilinear form $f(A,B)=\operatorname{tr} (A^t M B)$ where $A,B$ are two $n\times n$ matrices I have to find a necessary and sufficient condition (on $M$) for $f$ to be symmetric. I found out ...
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Kernel of bilinear form

It is written in book, I read: kernel of bilinear form is space consisting of vectors $y$, such: $$Ker(\alpha)=\{y\in V:\alpha(x,y)=0,\ \forall x\in V\}$$ Nice I get it, but then it is said, that ...
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Bilinear forms and operators

I am introducing myself to bilinear forms. I would like a little help to do this exercise: Let $f$ and $g$ two bilinear forms on $V$, with $f$ non-degenerated. Show that there is an unique operator ...
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Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
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Is there a convergence proof for ADMM applied to biconvex/bilinear problems?

I wonder if there is a local convergence proof for ADMM applied to biconvex problems? More specifically, my problem is as follows: $\text{minimize}_{x,y} f(x) + g(y) + \| y \circ Ax \|_2^2 $ , ...
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On non-degenerate bilinear forms on infinite dimensional vector spaces

For any non-degenerate bilinear form $(\cdot,\cdot)$ on a vector space $V$ and a linear functional $f$, there exists $v \in V$ such that $f(v)=(v,w)$ for all $w \in V$. It's easy in ...
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Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...