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

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0answers
44 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all $k$-...
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1answer
28 views

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 ...
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1answer
126 views

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 ...
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2answers
100 views

Graded tensor algebra

Given a finite dimensional $\mathbb R$-vectorspace $V$ we can make $$ T(V) := \bigoplus_{n=0}^\infty V^{\otimes n}. $$ Here $V^{\otimes n} = V \otimes \cdots \otimes V$. An element of $T(V)$ looks ...
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1answer
58 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb Z[...
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1answer
65 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...
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0answers
74 views

The canonical perspective on the Hodge star operator [closed]

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
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1answer
61 views

Exercise from Rotman, Advanced modern algebra , $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$

(i) Let $p$ be a prime,. Show that $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$ , where $\mathbb{Z}_p \oplus \mathbb{Z}_p$ si viewed as $\mathbb{Z}$-module ( with $\mathbb{Z}_p $ I mean $\...
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1answer
98 views

Differential forms and wedge product exercise.

Show that $$\omega \wedge v(\left <a_1,a_2,a_3 \right>,\left <b_1,b_2,b_3 \right>) = c_1 dx\wedge dy + c_2 dx\wedge dz + c_3 dy \wedge dz.$$ I wasn't given the form of $\omega$ or $v$. ...
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1answer
83 views

Unclear Construction of Basis for Tensor Product

My problem lies in page 363 of Steven Roman's Advanced Linear Algebra (Here's a link). The author says that for each ordered pair $(e_i,f_j)$ where $\left\{e_i\right\}_{i\in I},\left\{f_j\right\}_{j\...
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1answer
37 views

$\wedge^{2} \ (\mathbb{Q}/ \mathbb{Z}) = 0$

I have to prove that $$\wedge^{2} \ (\mathbb{Q}/ \mathbb{Z}) = 0$$ where $\wedge$ is the wedge product. Any hint ?
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1answer
34 views

inner product and orthonormal basis problem

Let $w_{1},...,w_{n}\in V$. Let $g_{ij}:=T(w_{i},w_{j})$, where $T(w_{i},w_{j})$ denotes the inner product. I want to show that $g_{ij}=\displaystyle\sum_{k=1}^{n}a_{ik}a_{kj}$. Hint: suppose that $...
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1answer
77 views

wedge product - distributivity over addition

Wedge and tensor algebra are very new concepts to me and I want to understand how to prove the following property of the wedge product: $$\omega\wedge(\eta_{1}+\eta_{2})=\omega\wedge\eta_{1}+\omega\...
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1answer
89 views

Operations in the exterior algebra. Multiplication in the direct sum of rings.

Let the exterior algebra $\Lambda(V)$ of a vector space $V$ over a field $K$ be the direct sum of the exterior powers $\Lambda^k(V),\quad k\in\overline{0,n}$. Then an element $x\in\Lambda(V)$ has the ...
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1answer
118 views

wedge product and tensor operation problem

Let $\{e_{1},\ldots,e_{n}\}$ be the usual basis for $\mathbb{R}^{n}$ and let $\{\varphi_{i},\ldots,\varphi_{n}\}$ be the dual basis. Show that $$\varphi_{i_{1}}\wedge\cdots\wedge \varphi_{i_{k}}(e_{...
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1answer
101 views

Is the wedge product surjective?

Is the wedge product $\wedge : \Lambda^{p}(V) \otimes \Lambda^{q}(V) \to \Lambda^{p+q}(V)$ surjective, for $V$ a real vector space of finite dimension? What dimension does its kernel have?
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1answer
33 views

How to show $\nu=dx_1\wedge\ldots \wedge dx_n$?

Let $\nu$ be the $n$-form in $\mathbb R^n$ satisfying $\nu(e_1, \ldots, e_n)=1$ where $\{e_1, \ldots, e_n\}$ is the canonical base of $\mathbb R^n$. Let $\displaystyle v_i=\sum_{j=1}^n a_{ij}e_i$. How ...
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1answer
65 views

Does the “bi” in bilinear and biorthogonal mean different things?

Does the "bi" in bilinear and biorthogonal mean different things? Bilinear seems to be linear from both left and right sides but biorthogonal means the product is zero sometimes instead of always?
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0answers
33 views

Symmetric tensors and duals

Let V be a finite dimensional vector space and consider $(Sym^n V^\vee)^\vee$ where $\vee$ denotes thedual, i.e homogenous polynomials in V of degree n. Consider as well $S_n(V)$, consisting of fixed ...
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1answer
167 views

Chain Rule to Compute Second Derivative

I was going through Marsden's book, Elementary Classical Analysis, and came across the following exercise in Chapter 6. It reads as follows: If $f: A \subset \mathbb{R}^n \to \mathbb{R}^m$ and $g: ...
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1answer
259 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
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1answer
78 views

Why can't $\partial X^i/\partial x^j$ be the components of a tensor field?

From Paul Renteln, "Manifolds, Tensors and Forms" in a chapter on tensor fields: Exercise 3.22 Not every object with indices is a tensor field. Let $X = X^i \partial / \partial x^i$ be a vector ...
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0answers
71 views

How to show the differential form $\nu$ satisfies $\nu(v_1, \ldots, v_n)=\det(a_{ij})$?

In $\mathbb R^n$ consider the differential form $\nu$ satisfying $\nu(e_1, \ldots, e_n)=1$. For every $i=1, \ldots, n$ consider the vector $\displaystyle v_i=\sum_{j=1}^n a_{ij} e_j$. How to show $\...
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1answer
112 views

Tensor powers of injective linear maps of free modules

This is a basic question on tensor products of linear maps. Let $R$ be a commutative ring and let $\varphi: M\to N$ be an injective linear map of finitely generated free $R$-modules. Question: Are ...
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0answers
46 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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2answers
86 views

Geometry of $k$-forms and $k$-vectors

In this question I was trying to see why $k$-forms are selected as the way to generalize vector calculus rather than $k$-vectors and a comment providing links to other questions made me end up with ...
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0answers
224 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
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2answers
72 views

Why there is this relation between $k$-vectors and $k$-forms?

I've been trying to understand the geometrical meaning of $k$-vectors and $k$-forms on some vector space $V$ of finite dimension $n$ over a field $\Bbb K$. Indeed, as I understood, a $k$-form $\omega \...
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1answer
105 views

linear independent or dependent set - linear algebra

I have the following set: $\{ [1; -1; -2], [-1;0;1], [1;2;1] \}$ and I need to find out whether the set is independent or dependent. My answer and the book's answer contradict. I thought it was ...
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0answers
69 views

Writing down proof about graded ideal in multilinear algebra

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
3
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2answers
452 views

Intuitively when to use the wedge product?

When I first learned the dot product and the cross product in $\mathbb{R}^3$ I spent some time understanding when I would like to use them. After some time I understood that the dot product usefulness ...
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1answer
156 views

Exterior power and alternating forms: explicit computations

I would like to get a more concrete understanding of a general isomorphism I have read about. I apologize if this is too basic, but I was not satisfied with the references at my disposal. Let $K$ be ...
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4answers
83 views

$\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$

Suppose $U$ and $W$ are $k$-vector spaces with bases $\{u_{i}\}_{i=1}^{n}$ and $\{w_{j}\}_{j=1}^{m}$. How to prove that $\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$ ?
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1answer
653 views

Linear Transformation induced by the following matrix A

Suppose $T:\mathbb R^4\rightarrow\mathbb R^4$ is the transformation induced by the following matrix $A$. Determine whether $T$ is one-to-one and/or onto. If it is not one-to-one, show this by ...
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1answer
118 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
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1answer
136 views

Tensor Einstein summation notation

I have two tensors $A^i$ and $B_j$ with components $(2,3,4)$ and $(1,2,3)$ respectively. What is the difference between $A^i B_i$ and $A^i B_j$? Is it just: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
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1answer
149 views

Is $(V_1\otimes\cdots\otimes V_k)^\ast \simeq V_1^\ast\otimes \cdots \otimes V_k^\ast$ true for infinite dimensional spaces?

Suppose $V_1,\dots,V_k$ are vector spaces of finite dimension. Then I could prove easily that $(V_1\otimes\cdots\otimes V_k)^\ast\simeq V_1^\ast\otimes\cdots\otimes V_k^\ast$. My proof was like that: ...
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0answers
55 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes \...
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How to show that $v_1\otimes\cdots\otimes v_k = 0$ if and only if at least one $v_i = 0$?

I'm trying to show that given vector spaces $V_1,\dots,V_k$ (not necessarily finite dimensional) over the same field $F$ then if $v_i\in V_i$ we have $v_1\otimes\cdots\otimes v_k = 0$ if and only if ...
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1answer
62 views

$b:V \times W \to \mathbb{R}$ bilinear. Show an induced $\phi:V \to W^*$ surjective.

Let $V$ and $W$ be finite dimensional vector spaces. Let $b:V \times W \to \mathbb{R}$ a bilinear map satisfying: $\forall v \in V. (\forall w \in W. b(v,w)=0) \implies v=0$, $\forall w \in W. (\...
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1answer
106 views

Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
2
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1answer
92 views

Notation in Bleecker's Gauge Theory and Variational Principles

In the proof of the theorem that there is a unique linear isomorphism $\star:\bigwedge^k(E)\to\bigwedge^{n-k}$ on p.4 in Bleecker's Gauge Theory and Variational Principles he says For $\gamma\in\...
2
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1answer
58 views

Number of Involutive Automorphisms on a Clifford Algebra

Let $V$ be a vector space with dimension $n$ and $q$ a quadratic form on $V$. How many involutive automorphisms are there in $\mathcal{Cl}(V,q)$?
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1answer
175 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes V^...
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0answers
215 views

Sum of the squares of the minors of a matrix with orthonormal column vectors = 1?

Let $A$ be an $m \times n$ ($n \leq m$) matrix with real entries and orthonormal column vectors. Claim: For $1 < k \leq n$, the sum of the squares of the $k\times k$ minors of $A$ is always $1$. ...
4
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1answer
491 views

Tensor basis change

I have a question regarding tensors and basis changes, however upon searching the web I've found an infinity of definitions for tensors so I'll have to give the one I know first: Given a vector space ...
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1answer
759 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E \...
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0answers
133 views

Second derivative dot product

I was wondering whether I am correct that the second derivative of the dot product is this: Let $f:X \times X \rightarrow \mathbb{K}$ $(x_1,x_2) \mapsto \langle x_1,x_2 \rangle$ Then we have $f'(x_1,...
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1answer
113 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. $4\...
4
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1answer
114 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...