G
Topic
First Poster
Last Poster
External Direct Sum
We2592 0
Q) 1. Let 
be external direct sum of vector spaces 
and 
over a field 
.let 
and 
show that
i)
and 
is subspaces.
ii)
Q)2. Suppose
. Let 
be the external direct sum of and . show that is isomorphic to under the correspondence 
I face some trouble to solve this problems help me for understanding.
thank you.
be external direct sum of vector spaces 
and 
over a field 
.let 
and 
show that
i)
and 
is subspaces.ii)
Q)2. Suppose
. Let 
be the external direct sum of and . show that is isomorphic to under the correspondence 
I face some trouble to solve this problems help me for understanding.
thank you.
0 replies
Then 
is
Putnam 1968 A6
sqrtX 11
N
by ohiorizzler1434
Source: Putnam 1968
Find all polynomials whose coefficients are all 
and whose roots are all real.
and whose roots are all real.
11 replies
Affine variety
YamoSky 1
N
by amplreneo
Let 
. Is it possible to equip 
with a finitely generated k-algebra with one generator such that make be an affine variety?
. Is it possible to equip 
with a finitely generated k-algebra with one generator such that make be an affine variety?
1 reply
Reducing the exponents for good
RobertRogo 0
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let be a ring with unity such that for every 
there exist 
such that 
. Prove that
a) If
then 
b) If there is an such that 
then the result from a) may no longer hold.
Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
be a ring with unity such that for every 
there exist 
such that 
. Prove thata) If
then 
b) If there is an
such that 
then the result from a) may no longer hold.Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
0 replies
Differential equations , Matrix theory
c00lb0y 3
N
by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
3 replies
The matrix in some degree is a scalar
FFA21 4
N
by FFA21
Source: MSU algebra olympiad 2025 P2
invertible, for an infinite number of 
:
Is it true that
such that 
is a scalar
4 replies
![\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}}
\, \mathrm{d}u
\]](http://latex.artofproblemsolving.com/9/1/1/91165e6b531233c56b799ed32ded982d57b6c87a.png)
hard number theory problem
danilorj 4
N
by c00lb0y
Let 
and 
be positive integers. Prove that
![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](//latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
is not a perfect square.
and 
be positive integers. Prove that![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](http://latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
is not a perfect square.4 replies
1 viewing
maximum dimention of non-singular subspace
FFA21 1
N
by alexheinis
Source: MSU algebra olympiad 2025 P1
We call a linear subspace in the space of square matrices non-singular if all matrices contained in it, except for the zero one, are non-singular. Find the maximum dimension of a non-singular subspace in the space of
a) complex
matrices
b) real
matrices
c) rational matrices
a) complex
matricesb) real
matricesc) rational
matrices1 reply
