All Questions
Tagged with vector-spaces tensor-products
233
questions
2
votes
3
answers
479
views
Two definitions of antisymmetrization of a tensor?
I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following:
...
- 21
0
votes
0
answers
74
views
Complexification of a vector space $V$
The tensor product $V \otimes \mathbb{C}$ is formed by taking the real vector space $V$ (where $\dim V=n$) and extending its scalars from $ \mathbb{R} $ to $ \mathbb{C} $. Elements in $ V \otimes \...
- 874
0
votes
1
answer
52
views
Intution behind $\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V, W)). $
I'm currently self-studying Tensor products and came across this result:
$$
\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V, W)).
$$
Whilst I can follow the proof algebraically, I can'...
- 107
2
votes
1
answer
114
views
An exercise in the first edition of Serge Lang's 'Linear Algebra.' What does this exercise mean? I cannot understand it at all?
I am currently reading the Japanese translation of the first edition of Serge Lang's 'Linear Algebra.'
The following exercise is in this book:
Let $V$ be a finite dimensional vector space over $K$.
...
- 1,138
2
votes
1
answer
75
views
How does the quotient construction of the tensor product imply the tensor product definition of two functions?
Let $V$ and $W$ be vector spaces. To define $V \otimes W$ as a quotient space, let $F$ be the free product of $V$ and $W$, and $R$ the set spanned by
$$(v_1 + v_2, w) - (v_1, w) - (v_2, w)\\
(v, w_1 + ...
- 6,275
0
votes
0
answers
29
views
Dimension of a tensor product following field extension
Let E/F be a field extension (assuming finite). Let V be an n-dimensional vector space over F. How to show that the dimension of E ⊗ V over E equals n? I've set up the preliminaries, such as writing a ...
- 11
0
votes
1
answer
61
views
Unsure of construction of tensor product from bases
I've trouble understanding the definition of tensor products from the bases of the spaces which the operations is applied
Given two vector spaces $V$ and $ W $ over the same field, with bases $ B_V $ ...
- 19
1
vote
1
answer
34
views
unicity of tensor product decomposition
For the sake of this question, a tensor product of two vector spaces $V$ and $W$ over a field $K$ is a couple $(T,h)$ where $T$ is a vector space over $K$ and $h:V\times W\to T$ is a bilinear map ...
0
votes
0
answers
96
views
What is the difference between the tensor product of a covector and a vector and a covector acting on a vector as a 1-form?
I believe I'm getting quite confused between the tensor product and the dual vector as a 1-form acting on a usual vector. Essentially, I'm struggling to distinguish the difference between $\epsilon^{i}...
- 13
1
vote
0
answers
52
views
Showing that for any $R$-modules $U,\ V,\ W$ we have that $U\otimes(V\oplus W)\cong(U\otimes V)\oplus (U\otimes W)$
As the headline says, I want to show that for any $R$-modules $U,\ V,\ W$ we have that $U\otimes(V\oplus W)\cong(U\otimes V)\oplus (U\otimes W)$. I know that there has been questions for this before, ...
- 381
2
votes
0
answers
48
views
Verification of Proof: Existence and Correctness of Linear Map h in Tensor Product Space
First some definitions:
$$ f: V_1 \rightarrow V_2 , linear$$
$$ g: W_1 \rightarrow W_2 , linear$$
$$ h': V_1 \times W_1 \rightarrow V_2 \times W_2 $$
$$ h: V_1 \otimes W_1 \rightarrow V_2 \otimes W_2 ,...
- 23
3
votes
0
answers
75
views
Induced $G$-module - question about notation.
In Algebra Vol. 2 the author, P. M. Cohn writes the following in chapter 7.7
Let $H$ be a subgroup of finite index $r$ in $G$ and consider a right $H$-module $U$. From it we can form a right $G$-...
- 853
1
vote
1
answer
45
views
Tensor Product of Homomophism of Arbitrary Vector Spaces
Suppose we have four arbitrary vector spaces $V_1$, $V_2$, $V_3$, and $V_4$.
Is it always the case that $Hom(H_1\otimes H_2,H_3\otimes H_4)=Hom(H_1,H_3)\otimes Hom(H_2,H_4)$?
This pretty easily ...
- 1,034
2
votes
1
answer
123
views
Tensor product as general composite space
I have a question regarding the tensor product between vector spaces. In general it only captures the composite state based on basis states of the corresponding vector spaces V and W. In order to get &...
- 409
0
votes
1
answer
148
views
Is the tensor product of two vector spaces the set of all bilinear forms on the Cartesian product of the vector spaces with entries reversed?
Let $E_1$ and $E_2$ be two finite dimensional vector spaces over $\mathbb{R}$. Then is it true that the tensor product $E_1 \otimes E_2$, is defined as the set of all bilinear maps from $E_2 \times ...
- 180