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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Null Bilinear Forms $x^T A y = 0$, where $A$ is square and full rank.

Let A be a full rank square matrix (A has no null space). When does $y^T A x = 0$ occur ? It could be that this problem is case-specific, so please find attached a document where x,y, and A take ...
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Is the matrix filled with the areas of pairwise intersections of disks in a plane always positive semidefinite?

Consider disks $s_1, \cdots, s_n$ in the plane and let $a_{ij}$ be the area of $s_i\cap s_j$. Is it true that for any real numbers $x_1,\cdots, x_n$ we have $$ \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0$$ ...
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Identification of a curious bilinear form

As I was solving a linear algebra exam, a wild bilinear form $f:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ appeared: $$f(x,y)=n\cdot\sum_{i=1}^n{\left(x_i-m_x\right)\left(y_i-m_y\right)}$$ ...
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Example of degenerate scalar product

Suppose I have a vector space $V$ over $\mathbb{R}$. A scalar product on $V$ is a symmetric bilinear form, so such that $\langle v,w\rangle = \langle w,v\rangle~ \forall v,w \in V$. To my ...
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Inner Product on $P_3(\mathbb{R})$ true/false

The question was to classify each statement as true or false with justification: Define $\langle f, g\rangle$ = $\int_0^1 f'(x)g(x) + f(x)g'(x) \,dx $. Then, $\langle , \rangle$ is an inner product ...
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$\det$ is the only multiplicative nonzero quadratic form on $\mathcal M_2(\Bbb R)$

Let $q$ a nonzero quadratic form on $\mathcal M_2(\Bbb R)$ verifying the relation $$\forall A,B\in\mathcal M_2(\Bbb R),\; q(AB)=q(A)q(B)$$ The question is to prove that $q=\det$. What I have tried ...
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Subspace $U\subset U^{\perp}$ has dimension at most $n/2$

We have $V$ a $\mathbb{R}$-vector space of finite dimension $n$ with non-degenerate bilinear form $\phi:V\times V\rightarrow \mathbb{R}$ and subspace $U$ of $V$ with $U\subset U^{\perp}$. How do I ...
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Bilinear and linear map, then zero map

If $b:V\times V\rightarrow W$ is both bilinear and linear, then it is the zero map. How do I prove this? What I tried: linear: $b((x,y)+(x',y'))=b(x,y)+b(x',y')$ and ...
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GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, ...
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Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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Using Lagrange's diagonalization on degenerate linear forms

Let $A=\begin{pmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{pmatrix}$ be a real matrix. Find an invertible matrix $P\in M_{3}(\mathbb{R})$ such that $P^TAP$ is diagonal ...
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Bilinear maps and non-degeneracy

Let $\mathbb{G}_0$ and $\mathbb{G}_1$ be two multiplicative cyclic groups of prime order $p$, $g$ a generator of $\mathbb{G}_0$ and $e$ a bilinear map, ...
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What is the meaning of the eigenvalues of the matrix representation of a bilinear form?

Given a bilinear form $B$ on some finite-dimensional vector space $V$, we can always represent $B$ by some matrix $A$ such that $B(v,w) = [v]^TA[w]$. Thus we could associate the eigenvalues of $A$ ...
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How to show that the vectors of $ker \phi$ are linearly indipendent to other vectors in a orthogonal basis with respect to $\phi$

How can I show that, given a symmetric bilinear form in a finite dimensional vector space $V$ $\phi : V\times V \rightarrow \mathbb{R}$ then the vectors of the basis of $ker \phi$ (which I assume that ...
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A quadratic form is positive-definite iff its set of isotropic vectors is trivial

Considering a quadratic form $Q$ in a finite dimensional vector space $V$ can I say that $\mathscr{I}=\big\{ \vec{o} \big\} \iff Q $ is definite positive ? Where $\mathscr{I}$ is the isotropic ...
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What is inside $W \cap W^{\bot_{\phi}}$?

I'm confused about this. Consider a bilinear symmetric form $\phi :V\times V\rightarrow \mathbb{R}$ and its associated quadratic form $Q$. Then considering a generic vector subspace $W\subseteq V$, ...
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Bilinear form is bounded

$\newcommand{\ints}{\mathbb{Z}}$ $\newcommand{\norm}[1]{ \lVert #1 \rVert }$ Given a bilinear form $a(x,y) = \sum_{n\in\ints}\sum_{m\in\ints}v_m x_{n-m} y_n$. $x \in \ell^2$, $y \in \ell^2$, $v \in ...
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Orthogonal basis with respect to $\phi$ must contain $ker \phi$?

Does a basis which is orthogonal with respect to a bilinear symmetric form $\phi$ necessarily contain the vectors of $ker \phi$ ? Since the vectors in $ker \phi$ are orthogonal to all the vectors in ...
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Explicit form of indefinite orthogonal group

Let $b$ denote the symmetric bilinear form $b(v,w) = v \cdot gw$ on $\mathbb R^{n+1}$ where $\cdot$ denotes the standard inner product on $\mathbb R^{n+1}$ and $g$ is the block diagonal matrix with a ...
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Bilinear forms on Lie algebras that vanish on commuting elements

Let $\mathfrak g$ be a finite dimensional real Lie algebra. If a bilinear map $A:\mathfrak g\times\mathfrak g\to\mathbb R$ vanishes on commuting elements (i.e. $[U,V]=0\implies A(U,V)=0$), is there a ...
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Linear transformation to bilinear form

Let there be $f:V\times V \rightarrow F$ over a finite vector space, which for a certain base $(w) = \{ w_1, w_2, ... , w_n \}$ has a nonsingular representing matrix (e.g. $[f]_w$ is nonsingular). ...
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Bilinear form over C [duplicate]

Let $f:V\times V \rightarrow \Bbb C$ be a bilinear form in a finite inner product space. Will there always be a single linear transformation $T:V\rightarrow V$ for which $f(v,u) = \langle Tv,u\rangle$ ...
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If a bilinear functional is continuous at $(0,0)$, then it is continuous everywhere

I need help with this exercise: Let X be a complex Banach space. A bilinear functional on $X\times X$ is a map $B: X \times X\rightarrow \mathbb{C}$ such that for all $x,y \in X$, the maps ...
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Why is this matrix skew-symmetric?

Consider Bilinear Forms, and two real $n \times n$ matrices $B$ and $\tilde B$. Suppose we have that $$(Bx,x) = (\tilde B x, x)$$ for all $x\in \mathbb R^n$. What can we can say about $B-\tilde ...
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Tricky problem regarding Bilinear Forms,

Here's another old Linear Algebra comprehensive exam question: Let A and B be two real $nxn$ matrices. Part 1) Show that if A and B are symmetric, then for any $\vec x \in R^n$, $$((A^2 + B^2)\vec ...
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Can a matrix be orthogonal with respect to a positive definite $\phi$?

I've a doubt on othogonal matrices. I know that an orthogonal matrix is a matrix $O$ such that $O^{T}O=O^{-1}O=I$ and also that $O$ has on the columns and on the rows the coordinates of the vectors ...
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Orthogonal basis with respect to a symmetric bilinear form $\phi$

I'm confused about orthogonal bases with respect to a symmetric bilinear form $\phi$. Consider the quadratic form $Q: \mathbb{R^{4}} \rightarrow \mathbb{R}$ defined as: ...
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Using Lax-Milgram for linear ODEs

Consider an Sturm-Liouville deferential equation as: $$Lu=(pu')'+qu$$ and differential equation as: $$Lu+f=0$$ where $u(a)=u(b)=0$. We can convert the problem into a Lax-Milgram form for $f\in C[a,b]$ ...
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Manipulation of skew-symmetric linear map

Let $\Delta$ be a skew-symmetric $n$-linear map. I have the following in my notes and I am having trouble seeing how it follows: $$ \Delta\left(\sum_{i=1}^n{e_i}, \sum_{i=1}^n{(e_i)} -e_2, ...
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Properties of non-zero symmetric bilinear form

Let $f$ be a non-zero symmetric bilinear form on $\Bbb R^3$. Suppose that there exist linear transformations $T_i:\Bbb R^3\to\Bbb R, i=1,2$ such that for all $\alpha,\beta\in\Bbb ...
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Defining a Euclidean Structure on a real vector space.

This comes from a homework question: For $\bf x, \bf y$ $\in \mathbb{R}^n$, put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n 2x_i^2 - 2\sum_{i=1}^{n-1}x_ix_{i+1}$. Show that the corresponding ...
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Existence of endomorphism $f$ such that $b(v,\phi)=\phi(f(v))$ for bilinear map $b$

Let $V$ be a finite dimensional vector space over a field $F$ and let $b:V\times V^*\rightarrow F$ be a bilinear map (with $V^*$ the dual of $V$). How do I show that there exists an endomorphism $f$ ...
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Is my proof that b(x,y) = <Bx,y>, using bilinearity, correct?

Show that for a bilinear skew-symmetric function, mapping $\mathbb R^n \times \mathbb R^n \to \mathbb R$, $b(x,y)=\langle Bx,y\rangle$, where $\langle x,y\rangle$ is the inner product $\sum_j ...
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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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Coercive bilinear form from Poisson equation with Robin boundary conditions

I need to find if the bilinear form arising from the following problem is coercive or not : $$-v\Delta u = f \quad \text{in } \Omega$$ $$v\frac{\partial u}{\partial n} +hu = 0 \quad \text{on } ...
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Prove that the bilinear form can be presented as a product of two linear forms

Let $ f:\Bbb R^3 \times\Bbb R^3 \to \Bbb R $ be a bilinear form such the the rank of $f$ is 1. Prove that $f$ can be presented as a product of the linear forms, such that: ...
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Bilinear maps and functions of the form $(x,y) \mapsto ux^2 + 2vxy + wy^2$

I was recently reading from the book "Vectors, Pure and Applied", $\S 16.1$ on bilinear forms. It begins In section 8.3 we discussed functions of the form $$(x,y) \mapsto ux^2 + 2vxy + wy^2$$ ...
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Representative Matrix for a Bilinear Form where $M=\left( \begin {matrix} 2 & 4 \\ 8 & 6\end {matrix}\right)$

I don't understand how to solve the following question, and maybe the basic idea of it. Let $V=M^R_{n \times n}$ and let $f: V \times V \to \Bbb R$ define as $f(A,B)=tr(A^tMB)$, for each $A,B \in ...
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Natural isomorphism between linear space to bilinear space

Let $V$ and $W$ be (not necessarily finite-dimensional) vector spaces. Show that there is a natural isomorphism (meaning an isomorphism that can be described without reference to a basis) between ...
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Image of a dual map is equal to Image$^{\perp\perp}$

Given two dual vector spaces $V$ and $V^*$ as well as linear maps $\phi: V \to V$ and $\phi^*: V^* \to V^*$ such that $\phi$ and $\phi^*$ are dual, i.e. there exists a non-degenerate bilinear form ...
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Find matrix of bilinear form on Lie algebra?

I have a bilinear form $$\sigma_V:L\times L\rightarrow \mathbb{k}$$ $$\sigma_V(x,y)=tr(\rho_V(x)\rho_V(y)) \forall x,y \in L$$ and am looking for the matrix of this form. I am in the algebra $$L= ...
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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 ...