0
votes
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 $k$? ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 $V\otimes_K L?$
0
votes
1answer
58 views

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 $C(V\times W)$ ...
2
votes
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 $F$ ?
2
votes
1answer
223 views

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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
31 views

Tensor calculus

I want to find the following, $\nabla \cdot (\rho u u)$, where $\rho$ is a scalar and $u$ a vector. I get $$ \partial_\alpha(\rho u_\beta u_\alpha) = \rho u_\beta \partial_\alpha u_\alpha + ...
1
vote
0answers
18 views

How to define the tensor product via an initial morphism? [duplicate]

I am just getting to know some category theory. My understanding of the "universal property" (based on this Wikipedia article) is that it is characterized in terms of an "initial morphism" or ...
2
votes
1answer
62 views

tensor product of a vector space and finite field

I want to know how we can interpret and define the tensor product of V as a vector space with a F as a finite field?
0
votes
0answers
62 views

Universal property of free spaces

In class we had developed the notion of a free space in preparation for our study of tensor product. There is one statement in my notes about the universal property of free spaces that has been ...
1
vote
2answers
58 views

Tensor Product problem.

Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$. I know the second ...
1
vote
1answer
67 views

Tensor product equivalent definitions

I'm studying tensor products right now and I've came across multiple definitions. The one I'm confused with is when we have vector spaces $V$ and $W$ and we define the tensor product as the quotient ...
2
votes
0answers
113 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
1
vote
1answer
121 views

Tensor product, wedge product, Hodge product, dyad, or what?

Suppose I have two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^3$. I can regard $\mathbf{u}$ as a $3 \times 1$ matrix, and $\mathbf{v}$ as a $1 \times 3$ ...
2
votes
2answers
180 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
0
votes
0answers
58 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
1
vote
1answer
272 views

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 ...
4
votes
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 ...
0
votes
0answers
113 views

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} & ...
6
votes
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 ...
0
votes
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.
1
vote
0answers
59 views

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?
7
votes
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?
4
votes
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 ...
4
votes
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 $d^1$ ...
1
vote
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 ...
3
votes
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 ...
2
votes
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 ...
7
votes
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 $\alpha : V \times W \rightarrow U$, ...
2
votes
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 ...
13
votes
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 ...
4
votes
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.