2
votes
1answer
43 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
0
votes
0answers
21 views

Finding this Polynomial Subspace

Let $A = k[x^{\pm 1}, y^{\pm 1} ] $, considered as a $k$ - algebra. Can someone give me a nice description of the (vector) subspace: $$ A_0 = \lbrace (f,g) \in A^2 : \frac{ \partial f}{\partial y} = ...
4
votes
0answers
28 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 ...
0
votes
0answers
30 views

When a system of rational linear homogeneous equations have complex solutions

Question: 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 ...
5
votes
1answer
64 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
61 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
95 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
62 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
38 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
39 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
21 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
43 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
votes
1answer
56 views

Abelian group over a field underlying an abstract vector space [closed]

Given that a set V is said to be a vector space over a field F if V is an Abelian group under addition and for each $a\in F$ and $\boldsymbol{v}$ in V there is an element $a\boldsymbol{v}$ in V, how ...
1
vote
1answer
36 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
37 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
54 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
26 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: ...
1
vote
2answers
55 views

Why is tensor product of linear maps defined as $(S\otimes T)(v\otimes w)=S(v)\otimes T(w)$?

In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the ...
-1
votes
1answer
32 views

Finding linear map given a condition.

Given that $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is linear and $T(3,2) =(4,6)$ and $T (2,3) =(1,-1)$ ,how can I find $T (4,3)$ ?.
2
votes
1answer
46 views

Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
0
votes
0answers
97 views

Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
1
vote
1answer
48 views

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
6
votes
1answer
59 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
0
votes
2answers
37 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
73 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
31 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 ...
1
vote
1answer
41 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
32 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
37 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
43 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
82 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
56 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
52 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
71 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
61 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
11 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
30 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
49 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
47 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
794 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
43 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
24 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
41 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
43 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?) ...