# Tagged Questions

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

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### 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
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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 ...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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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. $f(x) - f(p) =$ $\sum_j x_j^2 - \sum_k ... 2answers 86 views ### Geometry of$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 ... 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 ... 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 \... 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 ... 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 ... 2answers 461 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 ... 1answer 159 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 ... 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 ? 1answer 658 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 ... 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 ... 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 = ... 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: ... 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 \... 0answers 74 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 ... 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. (\...
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 $E$ ...