All Questions
Tagged with exact-sequence linear-algebra
58 questions
2
votes
1
answer
32
views
Relation between the elementary Rouché-Fröbenius theorem and the more abstract Fröbenius theorem with exact sequences.
In highschool I was taught that the "Rouché-Fröbenius theorem" was that given a homogenous linear system of equations, the solution space has dimension $n-r$ where $n$ is the dimension of ...
1
vote
0
answers
37
views
Exact sequence of subspace
Let $V$ be a finite dimensional vector space over $k$ and $V_1, V_2 \subseteq V$ be two subspace of $V$. Suppose that $W \subseteq V$ is a subspace such that $V_1, V_2 \subseteq W$.
We have two ...
- 103
-2
votes
2
answers
72
views
Having trouble finding an exact sequence. [closed]
Find an exact sequence: $\{0\} \rightarrow \mathbb{V} \xrightarrow{{\alpha}} \mathbb{U} \xrightarrow{{\phi}} W \rightarrow \{0\} $, such that $$\dim(\mathbb{V}) = \dim(\mathbb{U}) = \infty$$ $$\dim(\...
- 317
0
votes
0
answers
75
views
Short exact sequence of finitely generated R-modules
Assume that M',M'' are finitely generated R-modules (here R is a commutative unitary ring) and M is another R-module. The initial problem is that if there exists a s.e.s. $0\rightarrow M'\xrightarrow{...
- 394
0
votes
1
answer
54
views
Free chain complex and exactness
If $C_*$ is exact sequence of free $R$-modules, and $F:_R\text{Mod}\to _S\text{Mod}$ is additive functor, $FC_*$ is in general not exact. But if $R$ is a field, is $FC_*$ is exact?
(I think since $C_*$...
- 1,194
0
votes
0
answers
55
views
Commutative square inducing a linear transformation $r:\text{Coker }b\to \text{Coker }c.$
The following is based on an exercise from the book $linear algebra and geometry" by Leung.
$\color{Green}{Background:}$
Given the linear transformation $c:X\to Y$ and the commutative square ...
- 3,819
0
votes
0
answers
38
views
Meaning of the phrase "$X'$ can be identified with a subspace of $X$"?
The following is taken from
$\color{Green}{Background:}$
$\textbf{Exercise}$ Show that if a sequence
$$0\xrightarrow{}X'\xrightarrow{}X\xrightarrow{}X''\xrightarrow{}0$$
is exact, then $X'$ can be ...
- 3,819
0
votes
0
answers
29
views
How to show $\text{Im } i=\text{ker }\pi$ for sequece $0\xrightarrow{}X'\xrightarrow{i}X\xrightarrow{\pi}X/X'\xrightarrow{}0?$
The following is taken from "Linear Algebra and Geometry" by Leung.
$\color{Green}{Background:}$
$\textbf{Exercuse}$ Show that for each subspace $X'$ of a linear space $X,$ the sequence
$$0\...
- 3,819
0
votes
0
answers
63
views
Exact linear sequences of linear maps
Let
$ 0 \rightarrow V_1 \xrightarrow{\phi_1} V_2 \xrightarrow{\phi_2} V_3 \cdots \xrightarrow{\phi_n} V_n $
be an exact sequence of linear maps ($im(\phi_{i-1}) = ker (\phi_i)$ for all i. Show that ...
- 11
2
votes
1
answer
89
views
Equivariant splitting of short exact sequences with a $\mathbb {Z}/2\mathbb{Z}$-action
Let $G = \mathbb{Z}/2\mathbb{Z}$, and consider a short exact sequence of \emph{free} $\mathbb Z$-modules of finite rank endowed with a $G$-action
$$ 0\to \mathbb Z^n \to \mathbb Z^m \to \mathbb Z^k\to ...
- 5,960
1
vote
0
answers
107
views
Relationship between the dual of a subspace and its annihilator
Let $V$ be a finite-dimensional vector space and let $W$ be a subspace of $V$. I shall use the following notation: $V^*$ will be the dual space of $V$, $Wº$ will be the annihilator of $W$, that is the ...
- 181
1
vote
0
answers
78
views
A four-term sequence of vector spaces is exact if and only if the intermediate map is an isomorphism
This is Proposition 25.3(i) from Tu's book on manifolds:
The four-term sequence $0 \rightarrow A \xrightarrow{f} B \rightarrow 0$ of vector spaces is exact if and only if $f: A \rightarrow B$ is an ...
- 6,641
1
vote
1
answer
139
views
morphism between short exact sequences of vector spaces
I have recently been trying to learn some linear algebra, and have been following the book: Elements of Linear and Multilinear Algebra by John M. Erdman. The book states a theorem and has left the ...
- 11
0
votes
2
answers
111
views
On exact sequences of finite-dimensional inner product spaces
Consider the exact sequence $U \stackrel{f}{\to} V\stackrel{g}{\to}W$ of finite-dimensional inner product spaces.
Show that the sequence $W\stackrel{g^*}{\to}V\stackrel{f^*}{\to}U$ is exact. In other ...
2
votes
2
answers
56
views
Nilpotent Matrix And Sequence Properties
The original problem from algebra book asks to prove that if $A$ is a $2\times2$ nilpotent matix then $A^2=0$ . Can this be related with some properties of exact sequences ? I.m. that such matrix is a ...
- 111
0
votes
0
answers
96
views
Dual of short exact sequence of Lie algebras exact?
Let $\varphi:\mathfrak{g}\to\mathfrak{q}$ be a morphism of finite-dimensional Lie algebras over a field $K$. Define $\varphi^\vee:\mathfrak{q}^\vee\to\mathfrak{g}^\vee$, $l\mapsto l\circ\varphi$, ...
3
votes
1
answer
85
views
Set of complementary subspaces as an affine space
Let $V$ be a vector space and $W$ a fixed vector subspace of $V$. Denote by $\mathcal{E}$ the set of vector subspaces $U$ of $V$ such that $V=U\oplus W$. It is stated in the Examples section here that ...
- 3,367
2
votes
1
answer
113
views
On an exact sequence of complex vector spaces
Let us consider the following exact sequence of complex vector spaces:
$0 \to A \to B \to C \to D \to E \to.....\to 0$, where $E$ and the later vector spaces are not necessarily zero.
By rank nullity ...
- 961
0
votes
1
answer
42
views
why can $g$ be considered as a linear function?
Here is the question I am trying to understand its solution:
Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$
Here is the part I ...
user965463
2
votes
1
answer
252
views
Short exact sequences of filtered vector spaces are split
Let $k$ be a field. By a filtered vector space over $k$, I mean a pair $(V,F)$ where $V$ is a finite dimensional $k$-vector space and $F=(F^pV)_{p\in \mathbb{Z}}$ is an increasing filtration of $V$ by ...
- 1,548
0
votes
1
answer
75
views
Additivity with respect to a long exact sequence of
Suppose we have a long exact sequence of finite dimensional vector spaces
$$ \cdots\longrightarrow A^i\longrightarrow X^i \rightarrow B^i\rightarrow A^{i+1}\longrightarrow X^{i+1}\longrightarrow B^{i+...
- 11.3k
5
votes
0
answers
1k
views
Exact sequence induced by symmetric power of vector spaces
Let $k$ be a field, and suppose $0 \to M \to N \to P \to 0$ is an exact sequence of $k$-vector spaces. From this answer, we see that for any $r \in \mathbb{Z}_{>0}$ the map $$\mathrm{Sym}^r(N) \to \...
- 690
0
votes
1
answer
160
views
Linear algebra vs homological algebra
In linear algebra my book show the five lemma.
In wikipedia I noticed connection with the 9-lemma (=? 3x3 lemma), the snake lemma a.s.o.
In the wikipedia page this lemma are described as being part ...
- 25
0
votes
1
answer
43
views
Exact short sequence [closed]
Given a subspace $V_1$ of $V$, then
$$
\{0\} \xrightarrow{} V_1 \xrightarrow{i} V \xrightarrow{\pi_{V/ V_1}} V/V_1
\xrightarrow{} \{0\}
$$
is a exact short sequence.
Q: Is this something that I ...
- 25
1
vote
1
answer
451
views
What are the implications of a surjection between free modules?
My professor said the following in his lecture:
Suppose $M$ and $P$ are free modules and $\alpha: M\rightarrow P$ is a surjection. Then $\alpha$ splits and the kernel of $\alpha$ is a summand of $M$.
...
- 125
2
votes
0
answers
83
views
Exact sequence of vector spaces and sum of dimension
Let
$0\xrightarrow{}E_0\xrightarrow{u_0}E_1\xrightarrow{u_1}E_2\xrightarrow{}\ldots\xrightarrow{}E_{n-1}\xrightarrow{u_{n-1}}E_n\xrightarrow{u_n}0$
be an exact sequence. Then $$\sum_{2k+1\leq
n}\text{...
- 417
1
vote
1
answer
83
views
For a commutative exact sequence show that $V_1$ and $V_3$ are finite dimensional iff $V_2$ and $V_4$ also are.
For the following commutative exact sequence:
\begin{array}\\
&V_1 & \stackrel{A_1}{\longrightarrow} & V_2\\
& \uparrow_{A_4} &&\downarrow _{A_2}\\
&V_4 & \stackrel{...
- 197
2
votes
2
answers
93
views
How to show that $AA^*+B^*B$ is invertible if $U\stackrel{A}\to V\stackrel{B}\to W$ is exact?
Let $U, V,$ and $W$ be finite dimensional vector spaces with inner products. If $A: U \rightarrow V$ and $B: V \rightarrow W$ are linear maps with adjoints $A^{*}$ and $B^{*},$ define the linear map $...
- 1,754
0
votes
1
answer
64
views
Five term exact sequence - basis for the middle when I understand the rest?
Suppose I have an exact sequence of say finite dimensional $\mathbb{Z}/2$ vector spaces
$$
A \to B \to C \to D \to E
$$
and suppose I have bases for $A,B,D,E$ and I completely understand the maps $A \...
- 5,401
-2
votes
1
answer
83
views
Exact sequences (homomorphism of short exact sequences) [closed]
Provided is a homomorphism of short exact sequences:
$$
\begin{array}
& 0 & \rightarrow & F_1 & \overset{\varphi_1}{\rightarrow} & E_1 & \overset{\psi_1}{\rightarrow} & ...
- 205
2
votes
2
answers
285
views
Short exact sequence of vector spaces
Suppose $V$ is a finite-dimensional vector space, and $U,W$ its subspaces. Is it possible to construct the following short exact sequence? $$0\rightarrow U\cap W\xrightarrow{f}U\oplus W\xrightarrow{g}...
- 1,484
2
votes
0
answers
112
views
Split exact sequence of linear maps implies split exact sequence of linear hom maps
If the exact sequence of left $A$-modules
$$0\xrightarrow{}F'\xrightarrow{u}F\xrightarrow{v}F''\xrightarrow{}0$$
splits, the sequence
$$0\xrightarrow{}\text{Hom}(E,F')\xrightarrow{\bar{u}}\text{...
- 733
1
vote
0
answers
90
views
Split exact short sequence
If the given short exact sequence
$$ 0\to M'\xrightarrow{f}M\xrightarrow{g}M''\to0 $$
splits then why is $M$ isomorphic to the internal direct sum of $M'$ and $M''$?
I know that if this sequence ...
- 39
0
votes
1
answer
187
views
Commutative Diagram of Vector Spaces with Short Exact rows
I found the following question in an online book of Linear Algebra exercises (page 69)
All of the mathematical objects are vector spaces, and thus the maps are linear. It does not specify if the ...
- 1,256
0
votes
1
answer
41
views
If $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence, then $V = 0$.
Show that if $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence of vector spaces over a field $K$, then $V = 0$.
I know that $\ker(g) = \text{im}(f)$, but how can I get that $...
- 484
1
vote
0
answers
54
views
Compatible splitting of sub-exact sequence
We work with vector spaces over a field. We have the following commutative diagram with exact rows:
$\iota_1$ and $\rho_2$ are the canonical inclusion and projection, and they have, respectively, a ...
- 6,388
2
votes
1
answer
184
views
Complete reducibility of a Lie algebra $\mathfrak{g}$ and splitting of short exact sequence of $\mathfrak{g}$-modules.
Suppose that $\mathfrak{g}$ is a semisimple Lie algebra. By Weyl's theorem on complete reducibility, $\mathfrak{g}$ is completely reducible. Now my book says that the following is equivalent:
For ...
- 1,237
2
votes
1
answer
279
views
Splitting of Lie algebra extensions: why does this linear map exist?
Let $\mathfrak{b}$ and $\mathfrak{g}$ be finite dimensional Lie algebras and let $(\tilde{\mathfrak{g}},j,\phi)$ be a Lie algebra extension of $\mathfrak{g}$ by $\mathfrak{b}$, so we have a short ...
- 1,237
0
votes
2
answers
2k
views
Short exact sequence of vector spaces splits always
Let $0\rightarrow E \stackrel{i}{\longrightarrow} F \stackrel{p} {\longrightarrow} G \rightarrow 0$ be an exact sequence of vector spaces. I want to prove that this exact sequence splits, i.e. that ...
- 9,802
0
votes
1
answer
122
views
Show that $Ax=0$.
I need a hint to help me get started with this problem:
Given the sequence of homomorphisms, $\mathbb{Z}^{m}\to \mathbb{Z}^{n} \to M\to 0$, where $M=\mathbb{Z}^{n}/K$ and $K=im(\phi_{A})\subseteq \...
user482939
8
votes
2
answers
348
views
Does $V/(\ker f \cap \ker g) \cong \text{im} f +\text{im} g$?
Let $k$ be a field and $V$ be a finite dimensional $k$-vector space. Let $f$ and $g$ be two $k$-linear endomorphisms of $V$ such that $f\circ g=g\circ f$.
Do we have an isomorphism of $k$-vector ...
- 1,548
2
votes
0
answers
206
views
If sequence of vector spaces is exact then the sequence of duals is also exact
Induced Exact Sequence of Dual Spaces
I was able to prove the result for short exact sequences, but my instructor did not say short exact sequence, just sequence. So it is not given that the maps in ...
- 266
0
votes
0
answers
243
views
Inclusion and quotient mappings in exact sequences
I have problems with understanding the role of inclusion mappings and quotient mappings in an exact sequence:
Let $V$ be a vector space. Let $U\subseteq V$ be a subspace of $V$. Then, $0\rightarrow U\...
user522841
0
votes
1
answer
453
views
Linear transformation and Exact sequences
Let V be a vector space over the field F, and W $\subset$ V a subspace. Then the linear transformation T: V $\rightarrow$ V / W is well-defined, since v $\in$ [v].
I understand v $\in$ [v], but how ...
user522841
3
votes
1
answer
373
views
existence of an exact sequence of abelian groups
Can there be a exact sequence of abelian groups as follows:
$$0\to\mathbb{Z}\xrightarrow{g}\mathbb{Z}\oplus\mathbb{Z}\xrightarrow{h}\mathbb{Z}\to\mathbb{Z}\to 0?$$
So I have an error in reasoning: If ...
user421621
9
votes
4
answers
2k
views
What is a short exact sequence?
I'll just quote my book here so you can see the definitions I have:
Suppose that you are given a sequence of vector spaces $V_i$ and linear maps $\varphi_i: V_i\to V_{i+1}$ connecting them, as ...
- 101
2
votes
1
answer
160
views
Why solving linear equations is taking a quotient by some subspace?
Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations.
But this lecture notes suggest ...
- 360
2
votes
1
answer
195
views
Getting an isomorphism from a short exact sequence of inner product spaces
Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
- 5,488
4
votes
0
answers
330
views
Rank Nullity Theorem (Short Exact Sequences)
In the book Manifolds, Tensors and Forms, there is the following theorem:
We have a short exact sequence $0\rightarrow \ker T\xrightarrow{\iota} V\xrightarrow{T} W\rightarrow 0$ where $\iota$ is the ...
- 575
0
votes
1
answer
227
views
Why do linear splitting maps of Lie Algebra central extensions induce cocycles?
If we consider a central extension $\mathfrak h$ of a Lie algebra $\mathfrak{g}$ by the abelian $\mathfrak a$:
$$0 \longrightarrow \mathfrak a \longrightarrow \mathfrak h \stackrel{\pi}\...
- 22.2k