# Tagged Questions

0answers
34 views

### Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
1answer
47 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, ...
1answer
37 views

### Would a set of tensors be an algebraic group closed under some operation?

Could a set of tensors be known as an algebraic group or why would that not have a group properties? The reason I'm asking is to understand different tensors.
1answer
39 views

### Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
0answers
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1answer
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### When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
1answer
136 views

### How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
0answers
382 views

### What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
1answer
150 views

### Alternating tensors vs $p$-vectors

Is there a reason to differentiate between alternating tensors and $p$-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors? Thanks.
1answer
42 views

### If $T$ is a $k$-tensor and $S$ is an $l$-tensor, then $\text{Alt}(T \otimes S) = (-1)^{kl} \text{Alt}(S \otimes T)$

Could someone please help me with the following algebra question? I know it should be easy, but the textbook leaves the proof to the reader and I am having a hard time with it. Thank you in advance. ...
1answer
147 views

### Why would anti-symmetric (0,2) tensor be traceless?

As it is, why would anti-symmetric (0,2) tensor be traceless? Is it because trace should allow any variable for its indices?
0answers
88 views

2answers
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### Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...