0
votes
2answers
23 views

Projections $P$ and $Q$ such that $I-(P+Q)$ is invertible.

Let $P,Q$ be endomorphisms of a finite dimensional linear space, such that $P^2 = P$ and $Q^2 = Q$. If $I-(P+Q)$ is invertible, then $P$ and $Q$ has the same rank. The solution is that $rk(P) = ...
4
votes
1answer
61 views

$\mbox{Im }A\oplus \ker A^t = V$

Let $A:V\to V$ be an endomorphism of a finite dimensional linear space. It's easy to see that $\mbox{Im }A\cap \ker A^t = 0$. Because if $w = Av\in \ker A^t$, then $0 = \langle A^tAv,v\rangle = ...
1
vote
0answers
30 views

identifying $\mathbb H$$^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H$$^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
0
votes
1answer
39 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
2
votes
1answer
31 views

Action of GL$(2,\mathbb{R})$ on symmetric matrices

This is a problem from an old qualifier. Let GL$(2,\mathbb{R})$ act on SYM, the real symmetric 2x2 matrices, via $S \mapsto A^T SA$ for $A \in$ GL$(2,\mathbb{R})$ and $S \in$SYM. Show that each ...
-3
votes
2answers
32 views

If $F_3 =\mathbb{ Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? [closed]

If $F_3 = \mathbb{Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? Please help! $\mathbb{Z}=\{ \text{set of integers}\}$.
0
votes
2answers
44 views

Show that if K is a non-zero ideal of Z/mZ,

Show that if K is a non-zero ideal of Z/mZ, then K is the principal idea. Please help!
0
votes
1answer
36 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
-1
votes
1answer
41 views

Binomial Coefficients form Basis for Rational Polynomials

How would we show that the polynomials $c_n(x):=\dbinom{x}{n}$ form a basis for $\mathbb{Q}[x]$?
2
votes
0answers
30 views

How similarity transformation is related to coordinate transformation?

I know that every matrix can be transformed into its Jordan form using similarity transformation. But I wanted to know, this transformation is related to shifting of coordinate systems?
4
votes
2answers
81 views

What kind of algebraic structure is this

I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with ...
1
vote
2answers
55 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
2
votes
2answers
51 views

How to bring $5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$ to the form $\langle x',Ax' \rangle = 1$

How can I bring $$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$ to the form $$\langle x',Ax' \rangle = 1$$ where $x' = \alpha x + \beta$ where $\alpha \in \mathbb{R}^+$ and $\beta \in ...
1
vote
1answer
66 views

Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
1
vote
2answers
60 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
0
votes
0answers
10 views

Smith Normal Form of a subset of an integral basis

Take the standard basis of $\Bbb{Z}^4$, $\lbrace e_1,e_2,e_3,e_4 \rbrace$, and an arbitrary permutation $\sigma \in S_4$. Take three distinct elements (name them $a_1, a_2, a_3$) from the set $\lbrace ...
-1
votes
0answers
28 views

Group Structure on the set of matrices over finite field

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 $q^{n^2}$. You must be agreed with me that the set $M_n$ ...
-3
votes
0answers
48 views

Every polynomial of degree $\geq 1$ in $F[x]$ , $F$ a field, is irreducible or factors into a product of irreducible polynomials.

I am trying to prove the following: Every polynomial of degree $n\geq 1$ in $F[x]$, $F$ a field, is irreducible or factors into a product of irreducible polynomials. I don't understand fields ...
0
votes
1answer
44 views

Allow $2 \Bbb N$ to denote the even integers $> 0$.

Please help! Allow $2\Bbb N$ to denote the even integers $> 0$. Say $a \in 2\Bbb N$ is irreducible if there are no numbers $b, c \in 2\Bbb N$ so that $a = bc$. (1) Show that if $n$ is an odd ...
7
votes
3answers
781 views

Meaning of math symbol ~

Segment of Example: t = ... More usefully, we have: t ~ n*log(n) Note: ~ means "similarity" like in geometry, same shape but not same size. How is it interpreted here? Edit: yes, t depends on n ...
-1
votes
1answer
37 views

basis vectors of a 2D lattice plane in a 3D lattice

I know the basis vectors of the three-dimensional lattice $\Lambda = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3} \}$. I also know the equation of the plane in this 3D lattice, suppose $Ax + By + Cz = ...
0
votes
0answers
23 views

Smith normal forms and a math program

I am interested to know the Smith normal form of $4 \times 2$ matrices $M$: The two cases of my interests are: (1). $$M_1= \begin{pmatrix} 3 & 0\\ -5 & 4\\ 4 & -5\\ 0 & 3 ...
0
votes
0answers
36 views

Canonical Forms For Matrices

In the following paper by Wedderburn what are the restrictions on the field $\mathbb F$ or on the linear application $\varphi$ that the author refers to obtain the matrix B? ...
1
vote
1answer
40 views

How do I show that an endomorphism is self-adjoint if and only if $\langle u, Tu \rangle \in \mathbb{R}$ for all $u \in \mathbb{V}$

Let $$(V,\langle \cdot , \cdot \rangle)$$ be a complex vector space. Let $T \in \mathcal{L}(V)$ be an endomorphism. Now I want to show, that $T \in \mathcal{L}(V)$ is self-adjoint if and only if ...
2
votes
0answers
38 views

solving recurrence relations for functions with more than one variable

Is there a way to find formula for a function on more than one variable which is given by recurrence relation with some initial conditions? e.g.if one knows the value of f(n,p,l) for all p,l where ...
0
votes
2answers
24 views

Relation between the euclidean space and a set of functions.

Let $n$ be an integer. In what sense can $\mathbb{R}^n$ be seen as the collection of functions $\lbrace n\to \mathbb{R}\rbrace$? (-what is $n$ here?) And also, does this (bijection of sets, I guess?) ...
6
votes
1answer
98 views

Existence of $p \times p $ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
0
votes
1answer
60 views

Abstract algebra true or false answer check

Sorry about the giant picture file, but typing up this many questions on Latex would take forever. My attempts are below, I am fairly sure 16+ are right My answers: -1T- -2T- -3F- -4F- -5T- -6F- ...
1
vote
1answer
37 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
4
votes
1answer
84 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
3
votes
2answers
35 views

Basis and dim of the set of all $n\times n$ symmetric matrices.

An $n \times n$ square matrix $A$ is called symmetric if $A^T = A$ Show that the set of all $n \times n$ symmetric matrices, denoted $S$, is a subspace of $M_n(\mathbb{R})$. Give a basis for $S$ ...
1
vote
0answers
42 views

Is it true for algebras A,B,C, that $(A+B)\cap C = A\cap C+B\cap C$?

Let $A,B,C$ be subalgebras of some algebra $X$. I've managed to show, that $A\cap C+B\cap C\subseteq (A+B)\cap C$. If $x\in A\cap C+B\cap C$, then $x=a+b$, where $a\in A\cap C, b\in B\cap C$. Since ...
2
votes
3answers
41 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
0
votes
4answers
67 views

Showing some transformation is linear

Let $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ be an operation defined by $$T(a+bx+cx^2+dx^3) = a + dx + (a+d)x^2 +(b-c)x^3$$ Show that $T$ is linear What I have done so far is look at it like ...
2
votes
1answer
70 views

What is the number of distinct subgroups of the automorphism group of $\mathbf{F}_{3^{100}}$?

Let $G$ denote the group of all the automorphisms of the field $\mathbf{F}_{3^{100}}$ that consists of $3^{100}$ elements. What is the number of distinct subgroups of $G$?
0
votes
0answers
48 views

Scale invariance property of function

Consider a function $f_j$:$\mathbb{F}_p$ $\longrightarrow$ $\mathbb{F}_p$ where the set $\mathbb{F}_p$ is defined as $\{0,1,2,\dots,p-1\}$. Clearly there are $p^p$ possible maps. Here the index $j$ ...
1
vote
1answer
52 views

Proof that intersection of two subspaces is {0}

Given $V$ a K-vector space, and $E_1, E_2$ subspaces of V. If $B_1=\{v_1,\dots,v_m\}$ and $B_2=\{w_1,\dots,w_s\}$ are two basis of $E_1$ and $E_2$ and the vectors of the basis are linearly ...
0
votes
1answer
54 views

Preimage of a subspace

Given that $f:E \rightarrow F$ is a vector space homomorfism and $W \subset F$ a subspace, I want to prove $$W \cap \mathrm{im}(f)=\{0\} \implies f^{-1}[W]=\ker(f)\ ,$$ where $f^{-1}[W]$ is the ...
-1
votes
1answer
42 views

Ring of linear transformations modulo finite rank transformations [closed]

Let $ K $ be a field and $ V $ be a vector space of countable dimension (infinite) over $ K $, and let $ L = L (V) $ be the vector space of $ K $-linear transformations on $ V $. Let $ I $ be the ...
1
vote
2answers
57 views

Conjugates of the upper triangular matices

It's a shame...I want to give an explicit description of the set $\bigcap_{m \in GL_n(K)} mUm^{-1}$, $U$ being the upper triangular subgroup of $GL_n(K)$. It seems to be $K^\times I_n$ but I do not ...
1
vote
2answers
30 views

Quotient spaces and quotient groups: equivalence classes and cosets

(Throughout this post, I am talking about vector spaces.) I had the pleasure of doing Abstract Algebra two semesters early, however, I feel like some general context was lost in the process. While I ...
1
vote
2answers
94 views

Idempotent operators over the exterior algebra

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that $\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$, where $\{X,Y\}=(XY+YX)/2$.
6
votes
1answer
70 views

Proof behind $S^n\cong SO(n+1)/SO(n)$

I have been trying to understand the fact that $S^n \cong SO(n+1)/SO(n)$. I believe I have the intuition correct at this point; consider the case when $n=2$ as we have $S^2 \cong SO(3)/SO(2)$.: We ...
6
votes
3answers
74 views

$f: SL_2(\mathbb{R}) \to GL_4(\mathbb{R})$ show that $Im(f)=SL_4(\mathbb{R})$

I was struggling with the following problem (from linear algebra): Let $V$ be the vector space of the $2 \times 2$ matrices with real coefficients. Consider the action of the group ...
0
votes
1answer
52 views

difficulties with prooving: K is a vector space over Z/pZ

I am trying to solve the followong exercise: Given is K as a field with finitely many elements. i) show that K is a vector space over $\mathbb{F}_p:=\mathbb{Z/p\mathbb{Z}}$, for some special values ...
2
votes
3answers
39 views

Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
1
vote
0answers
39 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
0
votes
0answers
27 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
0
votes
1answer
61 views

Show that $f$ is diagonalizable

Given an endomorphism $f$ on the vector space on $\mathbb{R}$ of dimension $n$ such that $f(f(x))=3f(x)-2x$. Let $E_1=\ker(f-Id)$ and $E_2=\ker(f-2Id)$. Show that: 1.$E_1$ and $E_2$ form a direct ...
0
votes
3answers
51 views

If $f,g$ are two endomorphisms of $E$ such that $f(g(x))=g(f(x))$ and $g$ is nilpotent show that: $f$ is invertible => $f+g$ is invertible

If $f,g$ are two endomorphisms of E such that $ f(g(x))= g(f(x))$ and $g(x)$ is nilpotent show that: A) If $f(x)$ is invertible then $f+g$ is invertible too. B) If $f(x)+g(x)$ is invertible then ...