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

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2
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1answer
32 views

Dimension of Bil(V)

Let $V$ be a vector space of finite dimension $n$, and let $\operatorname{Bil}(V)$ be the vector space of all bilinear forms on $V$. In some notes by Keith Conrad, he says in an exercise that ...
2
votes
2answers
249 views

In Grassmann algebra a la Browne, why are vectors dependent if their wedge product vanishes?

I'm reading John Browne's Grassmann Algebra, Vol 1 : Foundations. Early on, he asserts without proof that if $x$ and $y$ are any two vectors in the underlying (real) vector space such that $x \wedge y ...
4
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0answers
137 views

Differentiable manifolds, Serge Lang

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
1
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1answer
129 views

3-d Matrices (as in, $A=[a_{ijk}]_{ijk}$) [duplicate]

In programming I've come across the idea of arrays containing arrays containing arrays etc., and as it's pretty intuitive to think of an array of arrays as a matrix, it seems like a reasonable idea to ...
0
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2answers
351 views

Solution to a system of symmetric equations

After applying the Lagrange multiplier method, I got the following system of equations, which is quite symmetric: $(x+y)^2 + (x+z)^2 = \frac{2}{3} \lambda x$ $(y+x)^2 + (y+z)^2 = \frac{2}{3} \lambda ...
2
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1answer
87 views

Help with algebraic manipulation to prove that $\Omega (M)$ constructed from $\Omega^1 (M)$ forms an algebra over $C^\infty (M)$

In Baez´s Gauge Theories, Knots and Gravity he states that the differential forms on a n dimensional manifold M, $\Omega (M) = {\bigoplus}_p \Omega^p (M)$, constructed from $\Omega^1(M)$ and the ...
2
votes
1answer
149 views

Nontrivial expansion of a multivariate power series in form of a single variable series?

I am trying to interpolate a function defined over a three-dimensional real space: $$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$ Let assume I have $N_1 N_2 N_3$ points in the space which form ...
1
vote
1answer
221 views

Intuition behind symmetric and antisymmetric tensors

I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
4
votes
1answer
140 views

Elements of $\wedge^2V$ expressible in the form $v_1\wedge v_2$

If $V$ is a complex vector space, then an element $w\in \wedge^2V$ is of the form $v_1\wedge v_2$ for some $v_1,v_2\in V$ iff $w\wedge w=0$ in $\wedge^4V$. Could anybody give some intuition/show why ...
3
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0answers
149 views

A basis of the symmetric power consisting of powers

Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It ...
1
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2answers
201 views

Ordered triples solution to system of equations

How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations? $$ \begin{align} x^2+y^2+z^2&=194 \\ x^2z^2+y^2z^2&=4225 \end{align} $$
4
votes
2answers
211 views

Doubt in argument in proof of universal property

I have a doubt regarding one argument used to prove that the tensor product indeed has the universal property, namely that for any multilinear map $f:V_1\times\cdots\times V_p\to W$ if the tensor ...
3
votes
1answer
271 views

Tensor Product universal property

Let $V,W$ be vector spaces and $T$ the following mapping: $$ \begin{align*} T:V\times W&\to V\otimes W\\ (v,w)&\mapsto v\otimes w \end{align*} $$ Then $(V\otimes W,T)$ Satisfies the universal ...
1
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1answer
563 views

Associativity of Tensor Product

I have a doubt on the associativity of the tensor product. I know that the tensor product of vector spaces is an associative operation up to a linear isomorphism and I'm just trying to prove that. My ...
0
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0answers
92 views

Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
3
votes
1answer
221 views

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 : ...
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0answers
81 views

'Injectivity' of a bilinear map restricted to the set it can generate starting from a given vector

Here is a problem I'm stuck with: Let $V$ be a vector space (on a field $F$) of finite dimension $n$, $v\in V$ and $\mu : V\times V \mapsto V$ a bilinear application determined by its action on the ...
1
vote
1answer
36 views

2-form associated with a skew map

Given a two-form $\omega\in \Lambda^2V$ for some (say finite dimensional) vector space $V$ we may associate with $\omega$ a skew map $f_{\omega}:V\rightarrow V^*$ given by $X\mapsto \iota_X\omega$, ...
10
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5answers
2k views

Why is the tensor product constructed in this way?

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
5
votes
5answers
6k views

Multivariate Taylor Expansion

I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and ...
7
votes
1answer
767 views

Kostrikin's Definition of Tensor Product

I'm having serious trouble to understand the definition of tensor products from Kostrikin's Linear Algebra and Geometry. Until now I've understood a tensor as a multilinear map from the cartesian ...
2
votes
1answer
66 views

Clarifying Theorem 4.11 of Lang's Algebra textbook.

Can someone more explicitly describe Theorem 4.11 in Algebra? Let $E$ be a module over a commutative ring $R$, and let $v_1,\dots,v_n$ be elements of $E$. Let $A=(a_{ij})$ be a matrix in $R$, and ...
0
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0answers
41 views

The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors

I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors. So ...
7
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1answer
1k views

tensor product with dual space

I will explain what I know, and then I will ask my question. Let $V$ and $W$ be vector spaces such that at least one is finite dimensional. In class, we showed that if either $V$ or $W$ is finite ...
2
votes
1answer
46 views

Homogenous Polynomial Functions and the Symbol of a Differential Operator

I have a trivial question concerning Lawson/Michelsohn's "Spin Geometry", Chapter III.§1. There, the symbol of a differential operator $P$ is defined to be a section $\sigma(P)$ in the bundle ...
5
votes
1answer
273 views

Multivariate Gaussian equivalent for a Gaussian integration identity.

For a one-dimensional x, $$\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}dx=\frac{1}{2}\int_{-\infty}^{\infty}e^{-x^{2}}dx$$ This can be shown through integration by parts. There is a good derivation of ...
1
vote
1answer
49 views

Dimension $V_1\times…\times V_n$

I have n Vector spaces $V_1,...,V_n$ and would like to show that $\dim(V_1\times...\times V_n)=dimV_1+...+dimV_n$ Is it possible to show this relation using somehow that the dimension of the tensor ...
1
vote
1answer
77 views

Bilinear extension of a map defined only on pairs of independent vectors

Let $V={\mathbb R}^d$ and $$ A=\bigg\lbrace (v_1,v_2) \in V \times V \bigg| \ v_1 \ \text{and} \ v_2 \ \text{are linearly independent} \bigg\rbrace $$ Consider the maps $f:A \to {\mathbb R}$ ...
2
votes
3answers
563 views

Any suggestions for abstract algebra-multilinear algebra books?

I want to read a little about these: The characteristic polynomial and minimal polynomial of a $T \in\mathrm{End}(V)$, or given a matrix $A$, finding the Jordan form and when can I say it is ...
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votes
1answer
95 views

Topology.Linear transformation

Let $T:V \rightarrow W$ be a linear transformation and $S \in L^k (W).$ Verify that $T^*(S^{\delta})= (T^* (S))^{\delta}, \delta \in S_k.$ Here is what I did, but unfortunately it is wrong. ...
3
votes
0answers
362 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
2
votes
1answer
59 views

Is there a smallest sub-Grassmann algebra containing a given vector in Grassmann algebra?

Let $V$ be a $d$-dimensional $\mathbb C$-vector space and the Grassmann algebra $$\mathcal G (V):=\bigoplus_{n=1}^d V^{\wedge n}$$ where $\wedge$ denotes the antisymmetric tensor product. I was ...
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0answers
30 views

Relation between symmetric powers and G-linear morphisms

My multilinear algebra is pretty bad, so I just wanted to check if I my intuition is correct: $$ Hom_{S_n}(V^{\otimes n}, V^{\otimes n}) \cong Sym^n(End(V)) $$ where V is a finite dimensional vector ...
1
vote
1answer
120 views

A basis for k-tensors

Let $V$ be a vector space of dimension $n$ with basis $e_1, \dots, e_n$. Let $a_1, \dots, a_n$ be the dual basis for $V^\ast$. Show that a basis for the space $L_k(V)$ of $k$-linear functions on $V$ ...
2
votes
2answers
175 views

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 ...
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0answers
84 views

Two questions about generalizing multilinear maps.

A multilinear map from the product $V_1\times\ldots\times V_n$ of vector spaces over the same field $K$ to another vector space $W$ over $K$ is a map $\phi$ such that if we fix vectors $$v_1\in ...
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0answers
44 views

Ring of invariants for the action of rotation groups in tensors.

Consider the component-wise action of the group $SO(p)\times SO(q)$ in the tensor product of two real vector spaces $S^2(R^p)\otimes R^q$. How to parametrize orbits of this action ? For $q=1$ we ...
3
votes
1answer
44 views

What is the rank of this linear map defined on big and abstract spaces.

Let $V$ be a finite-dimensional space, and let ${\cal L}(V)$ denote the space of all endomorphisms of $V$. For any $\phi \in {\cal L}({\cal L}(V))$, there is a unique bilinear map ${\cal L}(V) ...
3
votes
1answer
80 views

Describe the invariant bilinear maps on the linear group

Apologies if this is a stupid question ; it is at least a natural question. Let $V$ be a finite dimensional space over $\mathbb R$ or $\mathbb C$. Denote by ${\mathcal L}(V)$ the vector space of all ...
1
vote
1answer
356 views

Geometric meaning of Gram determinant

Let $v_1,v_2$ be vectors in $\mathbb{R}^4$. Let $M$ be the $2\times 4$ matrix with rows $v_1,v_2$ in this order. The Gram determinant of $M$ is defined as the determinant of the $2\times 2$ matrix ...
5
votes
2answers
749 views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...
8
votes
3answers
813 views

The determinant function is the only one satisfying the conditions

How can I prove that the determinant function satisfying the following properties is unique: $\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and ...
5
votes
0answers
134 views

Exterior algebra of a subspace

Let $E$ and $E^\star$ be two vector spaces in duality according to a (possibly symmetric) non-degenerate bilinear form $\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb{R}$. Let $F$ be a subspace ...
2
votes
1answer
325 views

A generalization of Lagrange identity

Let $k,n$ be positive integers, $k\le n$. Let $v_1,\cdots,v_k$ be vectors in $\mathbb{R}^n$. Let $M$ be the $k\times n$ matrix with rows $v_1,\cdots,v_k$ in this order. The Gram determinant of $M$ is ...
5
votes
1answer
222 views

Methods of Multilinear Algebra in Representation Theory

I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...
2
votes
1answer
110 views

3-dimensional array

I apologize if my question is ill posed as I am trying to grasp this material and poor choice of tagging such question. At the moment, I am taking an independent studies math class at my school. This ...
7
votes
2answers
726 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
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1answer
141 views

What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$?

Let $V$ denote an $n$-dimensional real vector space, and $\wedge^n$ denote the $n$-fold exterior product. What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$? In the book Introduction to ...
5
votes
1answer
102 views

General trace relation

Let $V$ be vector space $\dim V=N$, and $A\in End(V)$. Denote $$ \wedge^k A^m(\mathbf{v}_1\wedge\dots\wedge\mathbf{v}_k)=\sum_{s_1,\dots,s_k=0,1,\sum_j s_j=m} A^{s_1}\mathbf{v}_1\wedge\dots\wedge ...
4
votes
1answer
106 views

Trace of the multiplication operator

Let $V$ be vector space, $\dim V=N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}:\omega\to \mathbf{b}\wedge\omega$, where $\omega\in\wedge V$ ($\wedge V$ is the entire ...