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

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2
votes
1answer
60 views

“Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions

Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$ ...
3
votes
2answers
182 views

Exterior power “commutes” with direct sum

I know that for vector spaces $V, W$ over a field $K$, we have the following identity : $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ ...
3
votes
4answers
330 views

Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
1
vote
1answer
81 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
1
vote
1answer
69 views

Can someone what this notation means?

I don't understand what does $\phi_I$ mean The proof includes writing $\phi_I$ as a product of $\phi_{i_1}\phi_{i_2},\dots$, but it doesn't explain what the LHS really means
3
votes
1answer
57 views

Calculating a cokernel on wedge product

i have a question... maybe it is easy and im only doing some mistake. Given a surjective homomorphism $f\colon \mathbb{Z}^n \rightarrow \mathbb{Z}^m$, then its kernel $K_f$ is isomorphic to ...
1
vote
0answers
28 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 ...
1
vote
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 ...
1
vote
1answer
69 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 ...
1
vote
2answers
65 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 ...
0
votes
1answer
46 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 ...
1
vote
1answer
56 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, ...
1
vote
0answers
45 views

The canonical perspective on the Hodge star operator

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. ...
1
vote
1answer
49 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 ...
1
vote
1answer
62 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 ...
1
vote
1answer
44 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 ...
-2
votes
1answer
35 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 ?
0
votes
1answer
29 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 ...
0
votes
1answer
32 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: ...
0
votes
1answer
78 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 ...
2
votes
1answer
54 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 ...
0
votes
0answers
20 views
2
votes
1answer
76 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?
1
vote
1answer
29 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 ...
0
votes
1answer
46 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?
0
votes
0answers
25 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 ...
1
vote
1answer
80 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: ...
10
votes
1answer
185 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 ...
2
votes
1answer
60 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 ...
1
vote
0answers
58 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 ...
3
votes
1answer
82 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 ...
1
vote
0answers
39 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 ...
1
vote
2answers
56 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 ...
6
votes
0answers
205 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 ...
2
votes
2answers
60 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 ...
0
votes
1answer
63 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 ...
0
votes
0answers
49 views

How to show this is a graded ideal?

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
votes
2answers
214 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 ...
0
votes
1answer
66 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 ...
0
votes
4answers
71 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 $U \otimes W$ ?
0
votes
1answer
176 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 ...
7
votes
1answer
77 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 ...
1
vote
1answer
97 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: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
7
votes
0answers
77 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: ...
1
vote
0answers
46 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 ...
1
vote
1answer
69 views

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 ...
0
votes
1answer
51 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. ...
2
votes
1answer
66 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
votes
1answer
52 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 ...
2
votes
1answer
53 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 $\mathcal{Cl}(V,q)$?