# Tagged Questions

1answer
50 views

### Does $V\otimes_k K\cong W\otimes_k K$ imply $V\cong W$?

Let $V$ and $W$ be two $k$-vector spaces of the same dimension and $K/k$ any field extension. If $V\otimes_k K\cong W\otimes_k K$ as $K$-vector spaces then are $V$ and $W$ already isomorphic over $k$? ...
1answer
37 views

### On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
1answer
61 views

### Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
0answers
26 views

### Notation for a vector space: $(\mathbb{C}^\infty)^{\otimes L}$

In a paper, the authors use the notation $(\mathbb{C}^\infty)^{\otimes L}$, where $L$ is a constant, for a vector space, but they do not give a definition. They also implicitly introduce an inner ...
2answers
22 views

### restricting invertible maps to get new maps

For V and W as vector spaces, let we define V ⊗ W and suppose T be a invertible linear map from V ⊗ W to itself with special condition, I want to know whether there exist something like restricted ...
1answer
49 views

### Elementary problem about Tensor product and Kronecker product defined by linear map

I have some perplexities when I reading references about tensor product. My main question is: How to define the tensor product between two vectors? It is clearly to define the tensor product ...
0answers
31 views

### Tensor Product over a field

This question appeared in my exam and I could not solve it. Let $L$ be a field, $K$ subfield of $L$. Assume that dim$_K(L)$=$m$, and $V$ be a $L-$vector space amd dim$_L(V)=n$. If as usual ...
2answers
50 views

### Tensor product of a vector space and a field

Let $F$ be a field and $V$ a vector space of finite dimension $n$ over $F$. Let $\overline{F}$ be the algebraic closure of $F$. and let $\overline{V}=\overline{F}\otimes_F V$ the tensor product over ...
1answer
46 views

### Embed a vector space into a tensor product

If $V$ is a $K$-vector space and $L$ is a field extension of $K$ then why is the map $v \to v\otimes 1$ an embedding of $V$ into $V\otimes_K L?$
1answer
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### Tensor Product of Vector Spaces - Quotient Definition

I'm trying to figure out exactly what the tensor product of vector spaces is. This is what I understand so far: If $V, W$ are vector spaces over a field $R$ then the free vector space $C(V\times W)$ ...
2answers
48 views

### Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
1answer
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### What is the difference between Cartesian and Tensor product of two vector spaces

In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension ...
0answers
29 views

### What is the most generic algebraic structure for which we can define a tensor product? [duplicate]

We can define a tensor product of two vector spaces. But vector spaces are themselves modules and we can also define a tensor product of two modules. My question is the following: are modules the ...
2answers
53 views

### Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
0answers
28 views

### Checking injectivity of a map out of a tensor product: suffice to check the decomposable tensors?

In showing that the map $$\Phi: \hom(V,V')\otimes \hom(W,W') \to \hom(V\otimes W, V'\otimes W')\\ [ \Phi(\sigma \otimes \tau)](v\otimes w)=\sigma v \otimes \tau w$$ is injective, the proof in Roman's ...
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1answer
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### Tensor product and Direct product.

Is the direct product of two vector spaces (over the same field) just the tensor product of two vector spaces over $\mathbb{Z}$? Am I right in thinking that essentially we would use the tensor ...
2answers
359 views

### Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$

The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector ...
0answers
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### Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
3answers
242 views

### Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces? I am having a very hard time at digesting tensor products ...
0answers
56 views

### Complexification of Vector Spaces using Tensor Products

Can anyone recommend me some references concerning complexification of vector spaces using tensor product? I'm stuck on the tangent bundle complexification of smooth manifolds.
0answers
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### Operations on vector spaces

Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
2answers
215 views

### Operators on a Tensor Product Space

Suppose $V$ and $W$ are vector spaces over the same field. Is $\text{End}(V \otimes W)$ the same as $\text{End}(V) \otimes \text{End}(W)$? Are there special names for these spaces?
1answer
266 views

### Tensor product of real numbers over the rationals

How do I show that $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}\not\cong\mathbb{R}$ as $\mathbb{R}$-vector spaces? Possible approaches I can think of (but can't implement) are to show that this tensor ...
2answers
228 views

### Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
0answers
112 views

### Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$H = H_1 \otimes \cdots \otimes H_n,$$ and let $\mathcal{H}$ be a ...
1answer
195 views

### Proving something is the basis of a quotient space

Let $k$ be a field which does not have characteristic 2. Let $M$ be the free $k$-vector space generated by two elements $\{ c, x \}$. Let $T(M)$ be the tensor algebra of $M$ and let $I$ be the ideal ...
1answer
288 views

### Finding a bilinear map that differs on two given points.

Let $V,W$ be finite dimensional real vector spaces, and let $(v,w) \not= (x,y)$ be two points in $V \times W$. Is it possible to construct a bilinear map $\alpha: V \times W \to \mathbb{R}$ such that ...
1answer
496 views

### Showing that the dual space of bilinear maps $V \times W \to \mathbb{R}$ satisfies the tensor product property, for finite dimensional vector spaces.

Let $U,V$ and $W$ be finite dimensional vector spaces, and define $B$ to be the vector space of all bilinear maps $V \times W \to \mathbb{R}$. Given a bilinear map $\alpha : V \times W \rightarrow U$, ...
1answer
244 views

### Why is the tensor algebra of a vector space non-commutative?

I was just curious in describing the notion that the tensor algebra of a vector space (That is the direct sum of all spaces containing k-tensors for each k) need not be commutative because I am having ...
2answers
2k views

### Understanding isomorphic equivalences of tensor product

I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following ...
1answer
319 views

### Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.