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Questions tagged [multilinear-algebra]

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

A multilinear function is a function from $V_1\times V_2\times\dots \times V_n\to V$ where the $V_i$ and $V$ are all vector spaces, and the function is a linear function into $V$ when restricted to each of the $V_i$. This includes the special case of bilinear functions.

For example, the ordinary dot product in $\Bbb R^n=V$ is a multilinear function from $V\times V\to \Bbb R$.

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Linear map induced by bilinear maps

Suppose $f:X\times Y\rightarrow Z$ and $g:X\times Y\rightarrow W$ are bilinear maps in the category of vector spaces (say, real). Define the null space $N_1(f) :=\{x \in X: f(x,y)=0 \ \forall\, y\in Y\}$ and $N_2(f)$ analogously. If it's known that…
ADD
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Equivalent definitions of non-degeneracy conditions

Let $k$ be a field, $V,W$ be two vector spaces over $k$ and $\beta : V \times W \to k$ be a bilinear form. I have always seen non-degeneracy for $\beta$ over $V$ stated in the following way : $$\forall \,v \in V\setminus\{0\}\,\,,\,\, \exists w \in…
jeanmfischer
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possibly found a "counterexample" to a multilinear algebra problem

Edit: Exact Question. my question is b part $\phi:E\times F\to G$ be bilinear $\psi:E\times F\to H$ be bilinear Given $N_1(\phi)\subset N_1(\psi)$ and $N_2(\phi)\subset N_2(\psi)$ Show that there exist a linear function $f:G\to H$ such that…
Anvit
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Understanding Binet-Cauchy Identity

There is a famous equation called Binet-Cauchy identity which states that $$ \left(\sum_{i=1}^n a_i c_i\right) \left(\sum_{j=1}^n b_j d_j\right) = \left(\sum_{i=1}^n a_i d_i\right) \left(\sum_{j=1}^n b_j c_j\right) + \sum_{1 \le i \lt j \le n}…
Abhisek
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A problem on mutlilinear algebra

In Greub's book on multilinear algebra, a problem asked to show $B(E,F;G)$ is isomorphic to $L(E;L(F;G))$ where $B(E,F;G)$ denotes the bilinear mapping from $E*F$ to $G$ and $L(A;B)$ denotes linear mapping from $A$ to $B$. $E$, $F$ and $G$ are…
user45765
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Tensor exercise multilinear algebra

Determine which of the following are tensors on $\mathbb{R}^4$, and express those in terms of elementary tensors $$f(x,y,z) = 3x_1y_2z_3 - x_3 y_1 z_4.$$ The solution say My questions What does tensors on $\mathbb{R}^4$ mean? It means…
Lemon
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Product of Determinants in n-form definition

Watching Frederich Shullers "Lectures on the Geometric Anatomy of Physics" series, he defines the determinant of an Endomorphism $\phi$ as $$\det \phi = \frac{w(\phi(e_1),\ldots \phi(e_n))}{w(e_1, \ldots e_n)}$$ where $w$ is the volume form on some…
Kevin Guo
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Determinant function theorem

I am trying to prove the following theorem from the book of mathematical physics of Sadri Hassani Proposition 2.6.9: Let $\Delta$ be a determinant function in the N- Dimensional vector space V. Let $|v\rangle$ and $\{|v_{k}\rangle \}_{k=1}^{N}$ be…
CristhianMartinez08
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How to find a base for the set of multilinear functions?

Let $L(V_1,V_2,...,V_n;\mathbb{R})$ the set of multilinear functions where $V_i$ are finite vector fields. Find a basis for $L(V_1,V_2,...,V_n;\mathbb{R})$. I've proved that $L(V_1,V_2,...,V_n;\mathbb{R})$ is a vector field and for any $G, F \in…
sango
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Finding the inverse of a map from $\wedge^{k}\Bbb V^*\to \text{Hom}(\wedge^{n-k}\Bbb V,\wedge^n\Bbb V^*)$.

I am new to differential geometry and I have encountered a problem regarding $k$-forms and multilinear algebra. Let $\Bbb V$ be a vector space of dimension $n$ and let $0\leq k\leq n$. For any $\alpha\in\wedge^{k}\Bbb V^*$, consider the map from…
UserA
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Showing that a function is a tensor

I am trying to solve question 4 in Munkres Analysis on Manifolds section 26. The question is determine if the following is a tensor on $\mathbb{R}^4$ and express those that are in terms of the elementary tensors on $\mathbb{R}^4$. $f(x,y) = 3 x_1…
ackshooairy
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Hodge Star basis-independence proof

I am trying to solve a problem form a workbook I found, I apologize in advanced for the notations and my grammar, since english is not my first language I might have some grammatical errors and some notation differences in my question. Te exercise…
Bajo Fondo
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Proof a map is multilinear

Let $n \geqslant 2$. Define the map τ from $\mathbb{K}^n$ × ... × $\mathbb{K}^n$ (where the factor $\mathbb{K}^n$ appears $n$ times) to $\mathbb{K}$ by $$τ ((x_{1},i)^n _{i=1}, ...,(x_{n},i)^n _{i=1} ) := x_{1,1} + x_{2,2} + ... + x_{n,n}.$$ Is the…
J.banks
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$\wedge$-product, bilinear

I want to show, that the $\wedge$-product is bilinear. It is $\displaystyle (\omega^k\wedge\eta^l)(v_1,\dotso,v_{k+l}):=\frac{1}{k!l!} \sum_{\sigma\in S_{k+l}} \operatorname{sgn} (\sigma) \omega^k (v_1, \dotso, v_k) \eta(v_{k+1},\dotso,v_{k+l})$ I…
MathUnicorn
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Computing the Hodge-* for a Scalar

Let $(E,g)$ be a real oriented inner product space with orthonormal basis $(e_1, \dots, e_n)$ with corresponding dual basis $(e^1, \dots, e^n)$. Then, for any $\beta \in \Lambda^0(V) := \mathbb{R}$, how does one prove that $$ * \beta = \beta…
ItsNotObvious
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