2
votes
2answers
44 views

An elliptic element of $SL(2,\mathbb{R})$ conjugate to a rotation

Show that an elliptic element of $SL(2,\mathbb{R})$ is conjugate to a rotation. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As $|\text{tr}(A)|<2$ ...
6
votes
2answers
254 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
2
votes
3answers
32 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
0
votes
2answers
39 views

$\dim (U_1\cap U_2)\ge \dim U_1+\dim U_2-\dim V$

I'm reading the excellent and incredible well-written book: Algebraic Function Fields and Codes by Henning Stichtenoth. I don't remember this theorem in my linear algebra course, maybe this is a ...
6
votes
4answers
147 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
1
vote
2answers
56 views

$\dim B/A=\dim B-\dim A$?

If $A,B$ are two vector spaces over $k$ such that $B\subseteq A$, can I say $\dim B/A=\dim B-\dim A$? I need of this result to prove a theorem I'm working on. Thanks in advance
1
vote
1answer
16 views

A hyperplane in a $k$-algebra

Let there exist a nonsingular bilinear pairing $B:R×R→k$, where $R$ is a finite dimensional algebra over a field $k$, such that $B(xy,z)=B(x,yz)$ for all $x,y,z$ in $R$. Why the set $\{z∈R∶B(1,z)=0\}$ ...
0
votes
1answer
47 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
0
votes
1answer
36 views

Dimension of an algebra/vector space

Does the dimension of an algebra/vector space have any connection to Euclidian spacial dimensions, for all algebras/vector spaces? I know some algebras/vector spaces can be represented in ...
2
votes
3answers
49 views

When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
2
votes
1answer
33 views

Some wedge product computation

I want to calculate $w\wedge w$ for $w=\sum a_{ij} e_i\wedge e_j$ where the sum is over all $1\leqslant i<j\leqslant 4$. Is there a neat way to do it without writing all the terms out? What about ...
0
votes
1answer
43 views

Permutation as a product of transposition

I'm trying to figure out how the proof of the following theorem works: THEOREM: Every permutation is a product of transpositions. The proof is based on noetherian induction. I don't understand how it ...
2
votes
1answer
25 views

Infinite dimensional Vector Spaces and Bases of Quotients

Given a basis $\mathcal{B}$ of an infinite dimensional vector space $V$, will a quotient of $V$ by a subspace $I$ necessarily have a basis? Is there a way of obtaining it using $\mathcal{B}$? My ...
0
votes
0answers
15 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
1
vote
1answer
44 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
6
votes
0answers
228 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
1
vote
2answers
32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
1
vote
2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
0
votes
0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
9
votes
0answers
81 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what (if anything) can one say in ...
1
vote
2answers
83 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
1
vote
1answer
36 views

killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
1
vote
0answers
43 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
1
vote
1answer
43 views

Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...
1
vote
2answers
28 views

How to show that a null potent linear transformation is invertible

V is a K vector space and $ψ : V → V$ is a null potent linear transformation i.e. $ψ^N = 0$ for a certain $N ∈ N$. Prove that $Id_V − ψ$ est an invertible element in the ring $L(V, V )$. My assistant ...
2
votes
0answers
21 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
0
votes
1answer
49 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 $k$? ...
0
votes
0answers
21 views

Extending a trace on algebra to a trace of systems of algebras

Suppose, we have a trace $\tau$ on some algebra $\mathcal{A}$, i.e. $$\tau(aA+bB)=a\tau(A)+b\tau(B)\ \forall A,B\in\mathcal{A}, \forall a,b\in\mathbb{C}$$ The question rises, what are then the ...
1
vote
1answer
38 views

Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
1
vote
1answer
36 views

Determine if the following is a subspace and find its smallest possible subspace of $\mathbb{R}^3$

$U_k = \{(x_i)_{1≤i≤n} \in \mathbb{R}^n\ |\ x_k = 0\}$. Is this a vector subspace of $\mathbb{R}^n$? For $n = 3$, what is the smallest vector subspace of $\mathbb{R}^n$ that contains $U_1, U_2, U_3$. ...
4
votes
0answers
86 views

Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
3
votes
1answer
66 views

Is it coincidental that the definition of similar matrices seems to be the conjugate automorphism?

Two matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ for which $B = PAP^{-1}$. This definition seems presented without any motivation behind it. However, I did notice that it ...
0
votes
1answer
19 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
1
vote
2answers
46 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
2
votes
1answer
58 views

Canonical embedding into dual space?

How would one go about proving that there is no embedding of a vector space into it's dual that is independent of a choice of basis? Thanks
4
votes
0answers
36 views

co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
2
votes
1answer
145 views

When a system of rational linear equations have complex solutions does it have rational solutions?

Problem: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
8
votes
2answers
252 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
0
votes
2answers
74 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
6
votes
3answers
100 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
3
votes
2answers
69 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
1
vote
3answers
41 views

Show that the image of a linear transformation is equal to the kernel

Let $\phi$ be a linear transformation such that $\phi: V\to V$ We are given the following facts: $\dim(V) = 8$ $\dim(\mathrm{Im}(\phi)) = 4$ $\phi\circ\phi=0$ Show that $\mathrm{Im}(\phi) = \ker ...
0
votes
0answers
44 views

Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
1
vote
1answer
22 views

Determining a spanning set for $X/\bigcap_{i=1}^N \ker{\lambda_i}$, where each $\lambda_i$ is a linear functional on $X$

Let $X$ be a vector space over a field $K$. Suppose that $\{\lambda_i\}_{i=1}^N$ is a collection of linear functionals $\lambda_i : X \to K$. Let $W$ be the subspace $\{ x \in X \mid \lambda_i(x) = ...
0
votes
0answers
47 views

A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
1
vote
1answer
38 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
1
vote
1answer
39 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
1
vote
0answers
58 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
1
vote
1answer
28 views

Prove Basis for symmetric matrix.

**Let V be the vector subspace of M$_{2}$ ($\mathbb{R})$ consisting of all symmetric matrices, That is A$^{t}$ = A. 1) Show that $\clubsuit$= $\left\{ \left(\begin{array}{cc} 1 & -2\\ -2 & 1 ...
0
votes
1answer
22 views

$m=17 \cdot 23 = 391$. With an exponent $e=3$ and encrypted word is $c=21$. Decrpyting exponent $d=235$. Find $w$.

Say $m=17 \cdot 23 = 391$. With an exponent $e=3$ and encrypted word is $c=21$. Decrpyting exponent $d=235$. Find $w$, when $w \equiv c^{d} \pmod{m}$. So far I have split it up like this: ...