# 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.

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### Construction of a quadratic form under inequality constraints

I'm interested in the following question : Let $A \in \mathcal{S}_d(\mathbb{R})^{++}$ be a positive definite symmetric matrix. Let $z \in \mathbb{R}^d$ with $z \neq 0$ and $f \in \mathbb{R}^d$. Can ...
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### proof that the bilinear form $a(u,v)$ is coercive

I am trying to show that $a(u,v) = \int_{-1}^{1} \omega u_x v_x dx$ is a coercive and bounded bilinear form to apply the Lax Milgram Theorem. $\omega$ is defined as $\omega(x) := \sqrt{1-x^2}$. I ...
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### Set of isotropic vectors of generalized inner product space V is linear iff V over a field with char 2?

So, I've been learning linear algebra from a rather old book I really like (Andrzej Białynicki-Birula, "Geometria z algebrą liniową") that talks a bit about "generalized inner products" (that is, ...
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### Total derivative of $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$

I have been trying to find the Fréchet derivative of the following function: $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$ $(x \in \mathbb{R}^n, A \in\mathbb{R^{m \times n}})$. One possibility would ...
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### If $T$ is a symmetric bilinear form on vector space $V$, and let $U$ be a finite dimensional subspace of $V$, then $V=U+U^{\bot}$

Here is the full question: If $T$ is a symmetric bilinear form on vector space $V$, and let $U$ be a finite dimensional subspace of $V$, then $V=U+U^{\bot}$, where $U^{\bot}$ is the orthogonal ...
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### Approximation of inf-sup stable variational problems

Consider flowing exercise: Let V be Hilbert space and $A: V\times V \rightarrow \mathbb{R}$ be a symmetric, elliptic (with constant $\alpha_1$) and continuous (with constant $\alpha_2$) bilinear ...
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### Condition when $V=W+W^{\perp}$ for $\dim V<\infty$

Let $\mathbb{k}$ be a field. Let $V$ be a finite-dimensional $\mathbb{k}$-vector space. Let $\varphi:V\times V\to \mathbb{k}$ ($\operatorname{char}\mathbb{k}\neq 2$) is bilinear (symmetric or skew-...
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### Prove boundedness for this bilinear form

I have the following bilinear form $$a(u,v) = \int_0^1 \bigg(\frac{du}{dx}\frac{dv}{dx}+v\frac{du}{dx}+uv \bigg) \, dx$$ defined on $H^1_0(0,1)$ where $u(0)=v(0)=u(1)=v(1)=0$. How do I show that ...
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### Notations and Definitions clarification in the Algebraic Duality of $c_0$ and $\ell_1$

I have a total of $3$ questions, my purpose is to clear up notation in the following question, since they are not introduced in the text and I cannot seem to find the notations online Below, we ...
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### Tensor inner product vs Vector space inner product

How does the definition of inner product presented in this article: https://en.wikipedia.org/wiki/Inner_product_space relate to the definition presented in this video: https://www.youtube.com/watch?...
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### Weak form of heat equation with Neumann boundary conditions

I'm trying to wrap my head around the ways to approach the heat problem. This time, with Neumann boundary conditions. Now, I have the theory down, but the only example I found leaves me behind by ...
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### Bilinear forms on finite-dimensional space

Reading Quadratic functions in geometry, topology, and M-theory by M. J. Hopkins and I. Singer (https://projecteuclid.org/download/pdf_1/euclid.jdg/1143642908), I reckon some unclarity on p. 410: ...