Questions tagged [multilinear-algebra]

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

Filter by
Sorted by
Tagged with
1 vote
1 answer
18 views

I want to find $m,l,n$ and $k$ and $(A_1,B_1),(A_2,B_2)\in V\times W$ such that $A_1B_1+A_2B_2\neq A_3B_3$ for any $(A_3,B_3)\in V\times W$.

I am reading "Tensor Algebra" by Takeo Yokonuma (in Japanese). Problem 2 (on p. 329) Let $V,W$ and $U$ be vector spaces over $k$, where $k$ is a field. Let $\mathcal{L}(V,W;U)$ be the set ...
user avatar
  • 6,147
0 votes
0 answers
92 views

Pullback of a constant coefficient form

Let $\omega=\sum_{I}C_Idx_I$, where $I=(i_1,\dots,i_n)$ is a multi-index and $C_I$ constants, be an $n$-form in $\mathbb{R}^m$, with constant coefficients. Here $dx_I$ means $dx_{i_1}\wedge\dots\wedge ...
user avatar
  • 333
1 vote
0 answers
42 views

Tensor Product, proof of isomorphism

Let V and W be two vector spaces and $$\Phi: Hom_K(\land^rV,W)\rightarrow Alt_K^r(V,W)\\ f \mapsto [(v_1,....,v_r)\mapsto f(v_1\land \cdot \cdot \cdot \land v_r)]$$ I need to show that $\Phi$ is an ...
user avatar
4 votes
0 answers
90 views

Linear Algebra - Tensor Products

Let $V$ be a finite-dimensional vector space over $\Bbb Q$ and $\bigwedge^kV$ be the $k$-th antisymmetric power. Let $\{ v_1, \dots, v_n \}$ be a basis of V. Define $$\pi: V^{\otimes k} \to \bigwedge^...
user avatar
1 vote
1 answer
33 views

Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.

I have proved in what follows that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$ and I would appreciate if someone would check my proof and/or point out how to ...
user avatar
  • 3,102
1 vote
0 answers
21 views

Functoriality in Multilinear Algebra

So I just encountered the word functoriality in my script for linear algebra, in the chapter multilinear mappings. I don't really know what it means and after some research it turns out it's a term in ...
user avatar
  • 115
1 vote
1 answer
69 views

Solving a matrix equation involving tensors

While doing my research, I came across the following matrix equation in $W \in \mathbb{R}^{n \times d}$ that I could not solve. $$ \sum_{i=0}^{t} X_{i} W Y_{i} + X'_{i}WY'_{i} = Z $$ where $X_{i}, X'...
user avatar
0 votes
1 answer
34 views

Computing $\omega^n=\omega\cdots \omega$ ($n$ fold product), where $\omega=\sum\limits_{k=0}^{n}dx_idy_i$

I have proved the following statement and I would like to have some feedback on my proof (is it correct? can it be improved?). Thanks. Write the coordinates on $\mathbb{R}^{2n}$ as $(x_1,y_1,x_2,y_2,\...
user avatar
  • 3,102
0 votes
1 answer
54 views

How to calculate the wedge product of three terms

This question was asked in my assignment on Linear Algebra and I am struck on it because I am not very good in Wedge products . Question: Suppose the standard co-ordinate on $\mathbb{R}^3$ are x,y,...
user avatar
  • 1,545
1 vote
1 answer
67 views

The set of $d\vec{x}_I$ with $I$ increasing is a basis of the vector space $\Lambda^k(\mathbb{R}^n)^*$ of alternating multilinear functions

I am trying to prove that the set of $d\vec{x}_I$ with $I$ increasing is a basis of the vector space $\Lambda^k(\mathbb{R}^n)^*$ of alternating multilinear functions but I am not sure I have done so ...
user avatar
  • 3,102
0 votes
0 answers
22 views

Checking my understanding of Einstein summation convention

I have a limited understanding of the convention so I'd like to check. Given a (0,2) tensor or bilinear map $(e_{i}\otimes e_{j})=\zeta_{ij}$ with $i,j=1,2,...,n$, and two covectors $\Omega^{1}=\sum^{...
user avatar
0 votes
0 answers
55 views

Why differ the definition of the tensor product in algebraic and analysis books?

In algebraic books, Tensor product is defined using quotient space and in analysis books, tensor product is defined using the bilinear map and linear functional. Why is it defined in two ways? Thanks ...
user avatar
0 votes
1 answer
61 views

What is wrong with my reasoning regarding tensor products?

$\def\Rbb{\mathbf{R}}$ Let $F$ be a subfield of the field $K$ and let $V$ be an $n$-dimensional vector space over $F$. Then $K\otimes_FV\cong K^n$. Considering the case when $F=K=\Rbb$ and $V=\Rbb^1$, ...
user avatar
  • 1,131
2 votes
1 answer
55 views

Dimension of the symmetric\alternating k-tensor over an $n$-dimensional vector space.

I want to solve this question: Suppose $V$ is a vector of dimension $n$ over a field $F$ of characteristic not equal to 2. Calculate dim $Sym^{k}(V)$(the symmetric k tensor ). I know that $(Sym^k(V))^*...
user avatar
  • 1,163
0 votes
1 answer
59 views

How to prove that $B$ is non-degenerate? does the question has typo?

Here is the question I want to prove: Let $V = M_2(\mathbb{R})$ be the space of all $2\times 2$ matrices over $\mathbb{R}.$ Show that $$B(X,Y) = det(X+Y) - det(X) - det(Y)$$ Where $X,Y \in V,$ is a ...
user avatar
  • 1,163
0 votes
0 answers
30 views

K is an infinite field, f is a nonzero polynomial, prove $\exists \alpha_1,\cdots ,\alpha_n \> \in \mathbb{K}\>s.t. f(\alpha_1,\cdots ,\alpha_n)\neq0$

$\mathbb{K}$ is an infinite field,$f(x_1,\cdots,x_n)\in \mathbb{K}[x_1,\cdots,x_n]$ is a is a nonzero polynomial, prove that $\exists \alpha_1,\cdots ,\alpha_n \> \in \mathbb{K}\>s.t. f(\alpha_1,...
user avatar
  • 67
1 vote
0 answers
69 views

Uniqueness of Wedge Product

In Lee's smooth manifolds it is an exercise to show that the wedge product $\wedge \colon \Lambda^k(V)\times \Lambda^l(V) \to \Lambda^{k+l}(V)$ is the unique associative, bilinear and anticommutative ...
user avatar
  • 847
3 votes
1 answer
129 views

What is a top form?

I'm sorry for such a basic question, but I can't seem to find this term defined anywhere. I'm trying to learn analysis on manifolds, and google has not been any help in figuring out what this thing is....
user avatar
  • 306
0 votes
1 answer
59 views

Proof that pullback commutes with exterior derivative

I found what looks kind of like a proof that the pullback commutes with the exterior derivative, but it is so simple that I feel as if it is almost certainly wrong. Here is what I have: Let $\omega = ...
user avatar
  • 306
0 votes
0 answers
13 views

Extend tensor mode-$n$ product to multivariate functions

Given a tensor $\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times\cdots I_N}$ and a matrix $U\in\mathbb{R}^{U_n\times I_n}$, we can define the mode-$n$ product $\mathcal{Y}=\mathcal{X}\times_n U$ by $$ y_{...
user avatar
  • 439
0 votes
1 answer
93 views

Calculating simple pullbacks

I am having great difficulty calculating what seem to be very simple pullbacks. Right now, I am trying to integrate the 1-form defined by $\omega = dy$ parameterized by $f:[0,1] \rightarrow \mathbb{R}^...
user avatar
  • 306
1 vote
2 answers
72 views

What allows a bilinear form (output: field element) to also work as a linear map (output: vector)?

A bilinear form $B: V × V → K$, when the inputs are 2 vectors, has 1 element of their field as output. A linear map $L: V → W$, when the input is 1 vector, has 1 vector as output. I have seen cases ...
user avatar
0 votes
1 answer
65 views

Do all bilinear forms have a signature? [closed]

If yes: starting from any arbitrary bilinear form, what is the algorithm to calculate its signature? If no: what are the conditions necessary for a bilinear form in order to have a signature? Is there ...
user avatar
1 vote
0 answers
79 views

Matrix-rank nonincreasing unitary tensor operations

I have a multidimensional array $A_{ijkl}$ $\in\mathbb{C}^{m\times n\times o \times p}$ indexed by four integers $i,j,k,l$. I will call $i$ and $j$ the "left" indices, $j$ and $k$ the "...
user avatar
2 votes
0 answers
78 views

Generalization of "symmetric positive definite" to higher dimensions?

Suppose I have a positive-definite function $f:\mathbb{R}^n\to \mathbb{R}^1$. Then the Hessian $\nabla^2 f$ is symmetric positive definite and I can write $\nabla^2 f=QQ^T$ for some real-valued matrix ...
user avatar
0 votes
0 answers
46 views

A $3\times3$ matrix is multiplied by an unknown $3\times1$ matrix and if the result is $\mathbf{0}$ how do I compute the unknown matrix?

So my question is basically how to find the numerical values of $x_1$ $x_2$ and $x_3$ in the following equation. $$ \begin{bmatrix} -1 & -2 & -3 \\ -2 & 2 & -2 \\ 1 & 2 & 3 \\ \...
user avatar
  • 11
1 vote
1 answer
79 views

Solution to a linear equation involving a skew-symmetric tensor

Say that $S(\mathbf{x})$ is a skew-symmetric $k+1$-tensor, that is, $S_{i_0,...,i_a,...,i_b,...,i_{k}}(\mathbf{x})=-S_{i_0,...,i_b,...,i_a,...,i_{k}}(\mathbf{x})$ for $a,b=0,...,k$, then find $S(\...
user avatar
  • 56
1 vote
2 answers
46 views

Northcott Multilinear Algebra Universal Property Proof

Northcott Multilinear Algebra poses a problem. Consider R-modules $M_1, \ldots, M_p$, $M$ and $N$. Consider multilinear mapping $$ \psi: M_1 \times \ldots \times M_p \rightarrow N $$ Northcott calls ...
user avatar
2 votes
2 answers
56 views

Why in the union $\bigcup_{p \in U}T_p^*(\mathbb{R}^n)$ all of the sets $T_p^*(\mathbb{R}^n)$ are disjoint?

I am currently reading the An Introduction to Manifolds by Loring W.Tu (2nd edition, pp. 34), and as a novice to differential geometry and topology it is not quite obvious to me why in the union $\...
user avatar
  • 1,379
1 vote
0 answers
32 views

Invariant polynomial function on Lie algebras.

Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \...
user avatar
  • 107
1 vote
1 answer
57 views

Show that $\big\{A(v_{i_1}\otimes \cdots\otimes v_{i_k})\in V^{\wedge k}:i_1<\cdots<i_k \big\} $ is linearly independent

Let $V$ be vector space over $\mathbb{R}$ and $V^{\otimes k}$ be the $k$th tensor power of $V$. Denote by $S_k$ the set which contains exactly all the permutations of a set with $k$ elements. Using ...
user avatar
  • 679
1 vote
0 answers
68 views

The volume element determined by some inner product and orientation

While solving the problem 4-7 in Spivak's "Calculus on Manifolds", I came up with a hypothesis which can be helpful for solving the problem. The following is the problem which I'm trying to ...
user avatar
  • 23
3 votes
1 answer
156 views

Why does $\det A$ change sign when any $2$ columns of $A$ are interchanged?

I have tried to reason, using multilinear forms, the well-known fact that the determinant of a matrix $[A]$ changes its sign if any two columns of $[A]$ are interchanged. I am not confident if my ...
user avatar
  • 597
3 votes
1 answer
57 views

A positive semidefinite bilinear form must be degenerate?

If $\beta$ is a positive semidefinite (symmetric bilinear)/Hermitian form on $V$, i.e $\beta(v,v) \geq 0$ for all $v \in V$ with possibility that $\beta(v,v) = 0$ for $v \neq 0$. Then it satisfes the ...
user avatar
  • 329
0 votes
0 answers
40 views

Continuity of a basis of a family of quadratic forms

Let $p$ be a parameter in a k-sphere $\mathbb{S}^k$ and let $Q^p$ be a family of real quadratic forms over $\mathbb{R}^n$, which is continuous with respect to $p$. For every fixed $p$, there exists an ...
user avatar
0 votes
1 answer
90 views

On the coherence of the tensor product between vectors.

Note. All vector spaces are of finite dimension. The question concerns the coherence between the tensor product of vector spaces and the tensor product between vectors. But first let's introduce some ...
user avatar
2 votes
1 answer
39 views

How do you show that $e_{I}^{*}(e_{J}):=\delta_{I,J}$ is a basis for $(\Lambda_{k}V)^{*}$, $\Lambda_{k}V:= V^{\otimes k}/A$?

In Tu's book "Geometry" there is the following statement on page 171: I would like to show that $span\{e_{I}^{*} \}=(\Lambda_{k}V)^{*}$. For some reason I get completely confused by all the ...
user avatar
  • 754
0 votes
0 answers
16 views

Find the kernel of a strange map of quotients of vector space

Let $V$ be a vector space of dimension $n$ and let $W \subset V$ be a $k$-dimensional subspace. Consider the map $$ \varphi: (V/W) \otimes V \to \wedge^2(V/W) $$ which sends for $a,b \in V$ $$ (a+W) \...
user avatar
  • 402
0 votes
1 answer
49 views

I need to solved this linear algebra equation with an unambiguous solution

You know that $$\left[\begin{array}{rrr} a & b & b \\ b & c & -b\\ c & d & a \end{array}\right] *\left[\begin{array}{rrr} a \\ 1\\ b \end{array}\...
user avatar
2 votes
0 answers
36 views

How to prove this Proposition about determinant functions and linear composition?

Let $\Delta$be a determinant function of $N$ dim vector space $V$. Let $|v\rangle$and $\{|v_k \rangle \}^N_{k = 1} $ in vector space $V$. Proof the following eqatuion: $$\sum_{j=1}^N(-1)^{j-1}\Delta(|...
user avatar
  • 31
5 votes
0 answers
79 views

How to prove the following determinant identity

Prove: $$ \begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
user avatar
1 vote
1 answer
68 views

Embedding from exterior product to tensor product space

I came across the following: For a basis $\phi_1,\dots,\phi_n$ of $V$ there is a natural embedding $V^{\wedge n}\hookrightarrow V^{\otimes n}$ defined as $$(\phi_1\wedge \cdots \wedge \phi_n) \...
user avatar
0 votes
0 answers
34 views

triple cross product (for beginner)

I was reading triple vector product. I saw an expression like this : $${\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\...
user avatar
  • 83
4 votes
0 answers
87 views

Decomposition of Hodge Operator

Given a decomposition of a vector space $V \simeq U \oplus W$. Then as taking the exterior algebra preserves coproducts (it is left adjoint to the forgetful functor from graded-commutative graded ...
user avatar
  • 414
1 vote
1 answer
54 views

About the two induced linear maps by a bilinear map

Suppose that $V, W$ are $n$ dimensional vector spaces on a field $K$ and $b:V \times W \to K$ is a bilinear form. And let $\phi$ and $\psi$ be the maps defined by the folloing: $$\phi : V \to W^* ; v \...
user avatar
  • 49
3 votes
3 answers
58 views

Proving that $\dim(\bigwedge^k(V^*)) = \binom{n}{k}$ without constructing an explicit basis

Text: Discussion: I find this argument kind of hard to follow because an explicit basis is never constructed; the argument seems kind of indirect. I'm relatively comfortable with the first sentence ...
user avatar
  • 3,154
0 votes
0 answers
44 views

Reconstructing a cubic form

I am given a number of points $n_1,\ldots,n_\ell$ from an integer binary cubic form $ax^3+bx^2y+cxy^2+ey^3$ (without any information on what $x,y$ generated them). Can I reconstruct the form $(a,b,c,e)...
user avatar
  • 30.9k
2 votes
0 answers
27 views

Find a minimal subspace $W_{\alpha}$ in $V$ for a given $\alpha \in \bigwedge^{p}(V)$ for which $\alpha \in \bigwedge^{p}(W_{\alpha}).$

Let $V$ be a vector space of dimension $n$, and $1\leq p \leq n$, show that for every $\alpha \in \bigwedge^{p}(V)$ there exists a minimal subspace $W_{\alpha}$ in $V$ such that $\alpha \in \bigwedge^{...
user avatar
0 votes
1 answer
43 views

Existence of a function from a quotient

On "Introduction to Smooth Manifolds" by John M. Lee, at page 309 we're talking about multilinear algebra. There's a proposition on the Characteristic Property of the Tensor Product Space. ...
user avatar
1 vote
1 answer
83 views

Show that $v_1\wedge\dots\wedge v_k = x_1\wedge\dots\wedge x_k \implies \text{span}\{v_1,\dots, v_k\} = \text{span}\{x_1,\dots, x_k\}$

Let $V$ be an $n$-dimensional space and $v_1,\dots, v_k \in V$ are linearly independent. It is clear that if $x_1,\dots, x_k \in V$ have the same span as $v_1\dots v_k \in V$ then there is a scalar $t$...
user avatar
  • 3,854

1
2 3 4 5
24