Tagged Questions
1
vote
1answer
19 views
Invariant hermitian forms and irreducible representations
Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$.
Let $E:V^{\bigoplus ...
0
votes
1answer
24 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
0
votes
3answers
26 views
$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
0
votes
0answers
38 views
Linear Combinations of Irrational Numbers: An Analysis on Architecture
Under what condition(s) is
$$ k_1\omega_1+\cdots + k_n\omega_n=c,$$
where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$?
I'm essentially trying to show that this is the case only so ...
1
vote
1answer
39 views
$n$-linear form: An Interpretation
What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level?
EDIT:
I'm just trying to show that every $n$-linear alternating form on a vector ...
1
vote
0answers
25 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
0
votes
1answer
35 views
Bilinear Forms: An Initial Condition Proof
Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
3
votes
1answer
70 views
$AX=C$: An Inconsistent Linear Equation [duplicate]
Question:
Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C \in F^n$ such that
the system of linear ...
2
votes
0answers
21 views
Extending transvections/generating the symplectic group
The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
1
vote
2answers
74 views
Inconsistent System of Linear Equations
Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C ∈ F^n$ such that
the system of linear equations $AX = C$ is ...
1
vote
0answers
32 views
$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis.
I don't ...
0
votes
1answer
33 views
Normal matrices with orthogonal basis
we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
3
votes
2answers
62 views
Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$
So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
0
votes
2answers
93 views
$GL(n, \mathbb{C})$ is algebraically closed? [closed]
Let $GL(n,\mathbb{C})$ the group of non-singular matrices. Is it algebraically closed? For $GL(1,\mathbb{C})$ is it true; but if I take linear combinations of elements in $GL(n,\mathbb{C})$ with ...
1
vote
1answer
32 views
$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$
Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
2
votes
2answers
43 views
$AB = I_m \overset{?}{\implies} n\geq m$
Let $A ∈ M_{m×n}(F)$ and $B ∈ M_{n×m}(F)$ be two matrices such that $AB = I_m$. What should I be thinking to prove that $n ≥ m$?
3
votes
2answers
64 views
(sur/in)-jectivity
I'm having trouble showing this:
Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective ...
0
votes
1answer
51 views
What is a better way to state this?
Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective (respectively, surjective).
What is a ...
0
votes
0answers
76 views
Space-Function Cross-Element Interplay
Has anyone ever thought to show these results diagrammatically?
$$(W_1+W_2)^0 = W^0_1\cap W^0_2$$
$$(W_1\cap W_2)^0 = W^0_1+W^0_2$$
I mean with set nesting and overlap and containment and mappings ...
0
votes
1answer
62 views
$W^0$ is a subspace of $V^*$
If $W\subset V$ is a subspace of $V$, and $W^0=\{f\in V^* | f(v)=0~\forall~v\in W\}$, then how do I show that $W^0$ is a subspace of $V^*$?
4
votes
0answers
54 views
“Convex” polynomials
Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as:
UPDATED (case 0 was missing):
0) $p(x)=x$, i.e., ...
1
vote
2answers
34 views
Pointwise order on polynomials
I'm curious about the following problem. Given two polynomials $p,q:\mathbb{R}\rightarrow \mathbb{R}$ is it possible to determine automatically if $p(x)\leq q(x)$ for all $x\in\mathbb{R}$?
I assume, ...
0
votes
0answers
26 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
0
votes
1answer
79 views
$(U\circ T)^{*} = T^{*}\circ U^{*}$
Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
1
vote
1answer
34 views
Image of the projection map onto an irreducible module
Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
3
votes
2answers
48 views
Determine the cokernel of a linear transformation between $\mathbb Q$ vector spaces
Let $f:E\longmapsto V$ be a linear map between finite dimensional $\mathbb Q$-vector spaces with bases $\{e_1,\cdots,e_n\}$ and $\{v_1,\cdots,v_m\}$
Define $coker(f)$ to be the quotient vector space ...
3
votes
2answers
185 views
Advice: Modern vs. Classics
First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
5
votes
4answers
128 views
Difference between Ring and Algebra?
In mathematics, I want to know what is indeed the difference between a ring and an algebra? Thanks!
1
vote
1answer
55 views
A vector space V is an irreducible End(V)-module
Let $V$ be a nonzero vector space. I consider as a $\operatorname{End}(V)$-module. Then it is irreducible.
My thoughts:
Let $U$ be a nonzero submodule and $u\in U-\{0\}$. I want to show that $U=V$. ...
0
votes
1answer
37 views
$N(T^n) = N(T^{n+1})$
Let $T : V \rightarrow V$ be a linear map and $\dim(V ) < \infty$.
How can I show that there is an $n > 0$ such that $N(T^n) = N(T^{n+1})$?
Could someone here possibly help?
3
votes
1answer
55 views
Diagram Chase Argument: $\text{rank}(T)=\text{rank}([T]^{\scr{C}}_{\scr{B}})$
\begin{eqnarray}
\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{eqnarray}
Theorem Let $T:V\rightarrow W$ be a linear map with $\scr{B}$ and $\scr{C}$ bases of $V$ ...
4
votes
0answers
25 views
Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code
I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection.
Let $C$ be some extended ...
3
votes
3answers
113 views
Are Clifford groups very *non-commutative*?
Clifford groups seem to be very non-commutative by the relation \begin{equation}
\gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}.
\end{equation} But is it really so? Can we put this degree of ...
1
vote
0answers
54 views
Full Rank Matrix with a specific construction
Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$
$$Z=\begin{bmatrix}
w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
1
vote
1answer
49 views
bilinear form $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$, find ortogonal subspaces, that satisfy…
Define $F$ as bilinear form $M_n(\mathbb{R}) \text{ x } M_n(\mathbb{R}) \rightarrow \mathbb{R}$
$F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$
Prove, that $F$ is represented by ...
0
votes
2answers
71 views
A question about the determinant of matrices with integer entries
Motivated by some Physics backgrounds, let's consider the following group $GL_n(\mathbb Z)$ which consists of matrices satisfying some conditions.
Let $M_n(\mathbb Z)$ be the set(not a group) of ...
3
votes
0answers
34 views
Duality of $Z(G)$ and $[G,G]$ in representation?
This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group.
I was thinking about its manifestation in group ...
0
votes
1answer
33 views
Converting linear equation over $GF(2)$ to system of the equations
I'm working on the problem of solving systems of linear equations over $GF(2)$ using SAT-solver. There is a one step in the algorithm that I don't clearly understand. During this step I need to ...
1
vote
2answers
30 views
$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
Question
Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$?
Attempt
\begin{eqnarray}
T\circ ...
2
votes
0answers
32 views
Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?
Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation}
d_{\alpha}|\#G,
\end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
2
votes
0answers
17 views
Exceptional Simple Jordan Algebra Cross product
Does anyone happen to know of an explicit construction of the cross product on the exceptional simple Jordan algebra, or perhaps a reference?
Context: I'm trying to see if $(D^* a)X b + a X (D^*b) = ...
-7
votes
0answers
120 views
Linear Algebra Midterm Review: Part II [closed]
Just as the first and the third, I've completed all of these problems below, but I'd like to see how other people do them too. Any help is greatly appreciated.
My "solutions" are below:
The ...
-2
votes
2answers
35 views
$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
0
votes
1answer
45 views
Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$
Theorem to prove:
Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
2
votes
1answer
58 views
$[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ provided $T \circ T = T$ [closed]
Let $V$ be a finite-dimensional vector space and let $T:V \rightarrow V$ be a linear map such that $T \circ T = T$. How should one prove that there is a basis $\beta$ of $V$ such that
\begin{eqnarray}
...
1
vote
3answers
67 views
Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?
Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
7
votes
1answer
73 views
Show $SL(2,\mathbb{Z})$ written as finite product of elements of a particular form
Prove that any element of $SL(2,\mathbb{Z})$ can be represented by a finite product of matrices of the following form. $$\begin{pmatrix}1-ab & a^2\\ -b^2 & 1+ab\end{pmatrix}.$$
We are given ...
1
vote
1answer
35 views
Diagonalization over rings and the dimension of the cokernel of an endomorphism
So, I'm trying to prove the following:
Let $\mathcal{O}$ be a DVR, $M$ a finite-rank free $\mathcal{O}$-module, and $\varphi \colon M \to M$. Then $\dim_k (M/\varphi(M)) < \infty ...
0
votes
1answer
23 views
Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$
I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
4
votes
4answers
92 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
