Tagged Questions

1
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1answer
36 views

Intuition behind symmetric and antisymmetric tensors

I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
3
votes
1answer
66 views

Elements of $\wedge^2V$ expressible in the form $v_1\wedge v_2$

If $V$ is a complex vector space, then an element $w\in \wedge^2V$ is of the form $v_1\wedge v_2$ for some $v_1,v_2\in V$ iff $w\wedge w=0$ in $\wedge^4V$. Could anybody give some intuition/show why ...
4
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0answers
74 views

A basis of the symmetric power consisting of powers

Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It ...
2
votes
1answer
65 views

Tensor Product universal property

Let $V,W$ be vector spaces and $T$ the following mapping: $$ \begin{align*} T:V\times W&\to V\otimes W\\ (v,w)&\mapsto v\otimes w \end{align*} $$ Then $(V\otimes W,T)$ Satisfies the universal ...
1
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1answer
72 views

Associativity of Tensor Product

I have a doubt on the associativity of the tensor product. I know that the tensor product of vector spaces is an associative operation up to a linear isomorphism and I'm just trying to prove that. My ...
0
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0answers
41 views

Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
1
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0answers
36 views

'Injectivity' of a bilinear map restricted to the set it can generate starting from a unique vector

Here is a problem I'm stuck with: let $V$ be a vector space (on a field $F$) of finite dimension $n$, $v\in V$ and $\mu : V\times V \mapsto V$ a bilinear application. The bilinear application $\mu$ ...
1
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1answer
19 views

2-form associated with a skew map

Given a two-form $\omega\in \Lambda^2V$ for some (say finite dimensional) vector space $V$ we may associate with $\omega$ a skew map $f_{\omega}:V\rightarrow V^*$ given by $X\mapsto \iota_X\omega$, ...
0
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0answers
40 views

Factoring Systems of Polynomials

I was thinking sometime about solving multidimensional systems of polynomials in a way analgous to how polynomials are solved today that is: Given $p(x) = a_0 + a_1x + a_2x^2 .. a_Nx^N$ Use ...
3
votes
5answers
252 views

Motivation for the Tensor Product [duplicate]

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
3
votes
1answer
120 views

Kostrikin's Definition of Tensor Product

I'm having serious trouble to understand the definition of tensor products from Kostrikin's Linear Algebra and Geometry. Until now I've understood a tensor as a multilinear map from the cartesian ...
3
votes
1answer
111 views

tensor product with dual space

I will explain what I know, and then I will ask my question. Let $V$ and $W$ be vector spaces such that at least one is finite dimensional. In class, we showed that if either $V$ or $W$ is finite ...
1
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1answer
41 views

Dimension $V_1\times…\times V_n$

I have n Vector spaces $V_1,...,V_n$ and would like to show that $\dim(V_1\times...\times V_n)=dimV_1+...+dimV_n$ Is it possible to show this relation using somehow that the dimension of the tensor ...
2
votes
3answers
170 views

Any suggestions for abstract algebra-multilinear algebra books?

I want to read a little about these: The characteristic polynomial and minimal polynomial of a $T \in\mathrm{End}(V)$, or given a matrix $A$, finding the Jordan form and when can I say it is ...
-2
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1answer
82 views

Topology.Linear transformation

Let $T:V \rightarrow W$ be a linear transformation and $S \in L^k (W).$ Verify that $T^*(S^{\delta})= (T^* (S))^{\delta}, \delta \in S_k.$ Here is what I did, but unfortunately it is wrong. ...
3
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0answers
96 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
3
votes
1answer
38 views

What is the rank of this linear map defined on big and abstract spaces.

Let $V$ be a finite-dimensional space, and let ${\cal L}(V)$ denote the space of all endomorphisms of $V$. For any $\phi \in {\cal L}({\cal L}(V))$, there is a unique bilinear map ${\cal L}(V) ...
3
votes
1answer
60 views

Describe the invariant bilinear maps on the linear group

Apologies if this is a stupid question ; it is at least a natural question. Let $V$ be a finite dimensional space over $\mathbb R$ or $\mathbb C$. Denote by ${\mathcal L}(V)$ the vector space of all ...
8
votes
3answers
301 views

The determinant function is the only one satisfying the conditions

How can I prove that the determinant function satisfying the following properties is unique: $\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and ...
1
vote
1answer
173 views

A generalization of Lagrange identity

Let $k,n$ be positive integers, $k\le n$. Let $v_1,\cdots,v_k$ be vectors in $\mathbb{R}^n$. Let $M$ be the $k\times n$ matrix with rows $v_1,\cdots,v_k$ in this order. The Gram determinant of $M$ is ...
1
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1answer
48 views

3-dimensional array

I apologize if my question is ill posed as I am trying to grasp this material and poor choice of tagging such question. At the moment, I am taking an independent studies math class at my school. This ...
1
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1answer
99 views

What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$?

Let $V$ denote an $n$-dimensional real vector space, and $\wedge^n$ denote the $n$-fold exterior product. What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$? In the book Introduction to ...
3
votes
2answers
352 views

Abstract linear algebra, trilinear forms

Let $V$ be an 3-dimensional vector space over $\mathbb{R}$. Let $\Lambda^3V^*$ denote the space of alternating trilinear forms on $V$. Note: An alternating trilinear form on $V$ is a map $\omega: V ...
1
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2answers
119 views

The Vector Space over another Vector Space

Is it possible to consider a vector space over another vectorspace instead over a field as usual, where came into play that we need a field? And in such a vector space, the vector could be represented ...
0
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0answers
156 views

Difference between affine combination and linear combination for image dataset

Lets assume that we have an RGB image that has as you know 3 planes, so it is for each point $f(x,y)= \{R,G,B\}$ and we want to create $5^{3}$ linear combinations of the above values converting ...
5
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1answer
157 views

Kernel of the Tensor Product of a Linear Map with Itself

For two vector spaces, $V$ and $W$, and a map $f: V \to W$, it is clear that: $$ \ker(f) \otimes V + V \otimes \ker(f) \subseteq \ker(f \otimes f). $$ Does the opposite inclusion hold? If so, I'd ...
2
votes
1answer
28 views

is there a way to bound the following 2-norm?

Let $C$ be a three-dimensional tensor of dimensions $n \times n \times n$. Define: $[C(x,y)]_k = \sum_{i,j} C_{ijk} x_i y_j$, i.e. $C(x,y)$ is a vector of dimension $n$. Is there a way to bound the ...
0
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0answers
31 views

is there a way to multiply in the following tensor?

I have an $R^{n \times n \times n \times n}$ tensor that maps a matrix to another matrix, call it $K$. I also have the matrix $C = A \times B$ where $C,A,B \in \mathbb{R}^{n \times n}$ and $\times$ is ...
1
vote
1answer
142 views

Tensor Algebra of Tensor Algebra

Suppose $V$ is a vector space and $T(V)$ the tensor algebra of $V$. What happens if we take $T(T(V))$ that is the tensor algebra of the (vector space) $T(V)$? I 'guess' I heard that $T(T(V)) \simeq ...
0
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0answers
56 views

is the following correct for tensor products?

Let's say that I have a three dimensional tensor $A = ((B \times_1 C_1) \times_2 C_2) \times_3 C_3$. where $B$ is $n \times n \times n$ tensor, $C_i$ are $n \times n$ matrices and $A$, as a result, is ...
2
votes
2answers
56 views

is there a tensor that does the following?

I want a tensor (in the multi-linear algebra sense) which takes as an input a matrix $A$ of size $n \times n$ and returns as output an $n \times n$ matrix which is diagonal (zero off-diagonal), and on ...
0
votes
1answer
53 views

Tensor Product Question

For a finite dimensional vector space $V$, is it true that $\bigwedge^{n - 1}V \otimes V = \bigwedge^{n}V \oplus \ker(\bigwedge^{n - 1}V \otimes V \overset{\psi}{\rightarrow}\bigwedge^{n}V)$ where ...
3
votes
0answers
129 views

How to invert this function on matrices which involves the permanent?

I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The ...
5
votes
1answer
253 views

An Expression for the Wedge Product

For the question below, I have the following definitions and concepts in mind: The $k^{th}$ exterior power of a real vector space $V$, denoted $\Lambda^k(V)$ can be realized as the quotient of the ...
9
votes
1answer
232 views

Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers

Let $T$ be an endomorphism of a finite-dimensional vector space $V$. Let $$f(x)=x^n+c_1x^{n-1}+ \dots + c_n$$ be the characteristic polynomial of $T$. It is well known that ...
7
votes
2answers
286 views

Why is the following map an isomorphism between $Cl(V,\omega)$ and $\operatorname{End}(\Lambda(V))$?

Suppose you have a vector space $V$ of dimension $2n$. I know that there exists a basis $x_1,\dots,x_n,y_1,\dots,y_m$ such that $\omega(x_i,x_j)=\omega(y_i,y_j)=0$ and $\omega(x_i,y_j)=\delta_{ij}$, ...
0
votes
0answers
351 views

Gram-Schmidt orthonormalization with polynomials

I'm doing Gram-Schmidt orthonormalization on a sequence of polynomials on $\mathbb{R}[x]$ with $\tau(f,g) = \int_0^\infty f(x)g(x)e^{-x} dx$ and I've got down to a sequence of orthogonal polynomials ...
2
votes
2answers
131 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
3
votes
1answer
130 views

Tensors as linear combinations of pure tensors.

Let $V$ be an n-dimensional real vector space, consider the space $F(V^p)$ of real functions on the p-fold cartesian product $V^p$ and its subspace $(V^{*})^p$ of multilinear functions (i.e. covariant ...
11
votes
1answer
242 views

Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
0
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1answer
391 views

Hermitian matrices that commute

My question is: If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the ...
2
votes
1answer
86 views

Symmetric power and characters

Let $V$ be a 2 dimensional vector space over $\mathbb{C}$. Then $W := Sym^{n}(Sym^{m}V)$ is a representation of $GL(V)$. For $g \in GL(V)$, I consider $\chi_{W}(g)$. Let $x$ and $y$ denote the ...
6
votes
2answers
252 views

Tensor Decomposition

Consider a tensor product $$ V^{\otimes n} = \underbrace{V\otimes\cdots\otimes V}_{n} $$ where $V$ is a vector space over $\mathbb R$, $\dim V = m$ , hence $\dim V^{\otimes n} = m^n$ . So every $A ...
4
votes
1answer
256 views

Orthogonal Complements in Vector Spaces

If $V$ is a finite dimensional vector space over any field $F$, we define an inner product on $V$ as a map $\langle \,, \rangle\colon V\times V\rightarrow F$, satisfying, $\langle u,v+w\rangle ...
1
vote
0answers
44 views

Alternating forms tangential to a subspace.

Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. ...
1
vote
1answer
143 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
1
vote
2answers
246 views

Basis for tensor products

Suppose $V_1$ and $V_2$ are $k$-vector spaces with bases $(e_{i1})$ and $(e_{i2})$, respectively. I've seen the claim that the collection of elements of the form $e_{i1} \otimes e_{i2}$ forms a basis ...
2
votes
0answers
81 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
0
votes
2answers
147 views

Symmetric Linear Transformations with trivial kernels

Let $V$ be a vector space. Let $A$ be a symmetric bounded multi-linear operator from $V \times V \rightarrow \Bbb{R}$. Suppose that $A(v,v) \neq 0$ for all $v \in V \setminus \{0\}$. This let us ...
5
votes
2answers
211 views

Decomposition of product of exterior products

Suppose $V$ is a $n$-dimensional vector space. What is the kernel of $$\bigwedge^p V \otimes \bigwedge^q V\longrightarrow \bigwedge^{p+q} V$$ here $p+q \le n$.

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