It is currently Thu Jun 21, 2018 1:37 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Exact sequencePosted: Wed Feb 17, 2016 7:34 pm
 Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Let $\displaystyle{\left(\mathbb{K},+,\cdot\right)}$ be a field. Consider the exact sequence

$\displaystyle{\left\{0\right\}\to V_1\to V_2\to V_3\to \left\{0\right\}}$ consisted of $\displaystyle{\mathbb{K}}$ -

vector spaces of finite dimension. Show that this sequence splits.

Top

 Post subject: Re: Exact sequencePosted: Mon Mar 07, 2016 2:33 pm
 Team Member

Joined: Tue Nov 10, 2015 8:25 pm
Posts: 313
To show that the given short exact sequence of finite-dimensional vector spaces splits, we have to show that $Im(f)$ is a direct summand of $V_{2}$, where $f \ \colon V_{1} \longrightarrow V_{2}$ is the given $\mathbb{K}$-linear monomorphism.

Since $f$ is a $\mathbb{K}$-linear map, $Im(f)$ is a vector subspace of $V_{2}$. Since $V_{2}$ is a finite-dimensional vector space, say of dimension $n$, $Im(f)$ is also finite-dimensional, say of dimension $k$. Therefore, consider a $\mathbb{K}$-basis $\{ e_{i} \}_{i=1}^{k}$ of $Im(f)$ and extend it into a $\mathbb{K}$-basis of $V_{2}$. It follows that $Im(f)$ is a direct summand of $V_{2}$, as every element $\displaystyle v$ of $V_{2}$ is expressed uniquely as $\displaystyle v = \sum_{i=1}^{k} \lambda_{i}e_{i} + \sum_{i=k+1}^{n} \lambda_{i}e_{i}$and the first summand belongs to $Im(f)$.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta