# Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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### Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? [duplicate]

Let $U$ and $V$ be modules over a commutative ring $K$. Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? I'm a differential geometer so I'm usually dealing with finite-...
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### Bivector to pseudovector mapping

I am studying antisymmetric tensors and currently reading a topic on pseudovectors. I understand that every bivector can be mapped to a corresponding pseudovector and vice versa but it is mentioned ...
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### From vec-trick to matrix-trick for Kronecker products

for the vec-trick of the Kronecker product, we can write $$\left(\mathbf{B}^{\top} \otimes \mathbf{A}\right) \operatorname{vec}(\mathbf{X})=\operatorname{vec}(\mathbf{A} \mathbf{X} \mathbf{B}).$$ ...
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Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\... 1answer 56 views ### If$\mathbb Q \otimes_\mathbb Z \mathbb Q \cong \mathbb Q^\mathbb N$, why is$\mathbb Q \otimes_\mathbb Z \mathbb Q$a$1$-dim$\mathbb Q$-v.s. In Dummit & Foote, it is an exercise to show that$\mathbb Q \otimes_\mathbb Z \mathbb Q$is a$1$-dimensional$\mathbb Q$-vector space. This is fairly easy: a$\mathbb Q$-basis for$\mathbb Q \...
Suppose $X \subset \mathbb P^n$ and $Y \subset \mathbb P^m$ are projective varieties, and let $S(X)$ and $S(Y)$ be their homogeneous coordinate rings. Consider the projective variety $X \times Y$ in \$\...