# 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.

1,157 questions
Filter by
Sorted by
Tagged with
10 views

1 vote
47 views

### $f \mapsto b, b(r,m)=r\cdot f(m)$ gives an isomorphism of $R$-modules $\Phi_N : \text {Hom}_R(M,N) \rightarrow \text {Bilin}_R(R×M,N)$

Let $R$ be a commutative ring $M$ and $N$: $R$-modules Let $f : M \rightarrow N$ is a homomorphism, I have shown that the map $b: R\times M \rightarrow N$ given by $b(r,m) = r \cdot f(m)$ is R-...
• 585
143 views

### Double Orthogonal Complements in a Bilinear Form with a Specific Matrix

I'm studying a problem from linear algebra: For a bilinear form $b: V \times V$ and a subvector space $U$ in $V$ we set $U^{\perp}:=\{v \in V \mid b(u, v)=0 \text { for all } u \in U\} .$ Now let ...
1 vote
19 views

### Fraction of bilinear forms vs bilinear form of fraction and the formulation of the objective function of Linear Discriminant Analysis

Let $A, B \in M(n, \mathbb{R})$ be $n \times n$ symmetric square matrices with real values and suppose $B$ is invertible. We can define three bilinear forms $(u,v) \mapsto u^TAv$ $(u,v) \mapsto u^TBv$...
• 503
71 views

• 711
1 vote
137 views

• 205
1 vote
49 views

### Let $f:V \times W \to \mathbb F$ and $s:V \to W^*, r:W \to V^*$. Find $[s]^B_{C^*}, [r]^C_{B^*}$.

Let $f:V \times W \to \mathbb F$ bilinear map and $s:V \to W^*, r:W \to V^*$ linear transformations s.t: $s(v)(w) = f(v,w)$ and $r(w)(v) = f(v,w), \forall v \in V$ and $\forall w \in W$. Let $B$ an ...
61 views

### Showing a bilinear form is coercive and continuous

Let $\Omega \subset \mathbb{R}^3$ be open and bounded. Define $H$ to be the completion of the space divergence-free smooth functions with compact support in $\Omega$ in the $(L^2(\Omega))^3$ norm. ...
• 5,539
1 vote