Tagged Questions

32 views

My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
71 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 $|\text{tr}(A)|<2$ ...
259 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 ...
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 ...
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 ...
150 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: ...
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
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 $\{z∈R∶B(1,z)=0\}$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
228 views

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 ...
41 views

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 $\mathrm{Im}(\phi) = \ker ... 0answers 44 views Defining an inner abstract vector space Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ... 1answer 22 views Determining a spanning set for$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) = ...
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 ...