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Putnam 2013 A5
Kent Merryfield   10
N Yesterday at 10:00 PM by blackbluecar
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be area definite for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
10 replies
Kent Merryfield
Dec 9, 2013
blackbluecar
Yesterday at 10:00 PM
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Linear algebra problem
Feynmann123   1
N May 25, 2025 by Etkan
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
1 reply
Feynmann123
May 25, 2025
Etkan
May 25, 2025
J
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External Direct Sum
We2592   1
N May 22, 2025 by Acridian9
Q) 1. Let $V$ be external direct sum of vector spaces $U$ and $W$ over a field $\mathbb{F}$.let $\hat{U}={\{(u,0):u\in U\}}$ and $\hat{W}={\{(0,w):w\in W\}}$
show that
i) $\hat{U}$ and $\hat{W}$ is subspaces.
ii)$V=\hat{U}\oplus\hat{W}$

Q)2. Suppose $V=U+W$. Let $\hat{V}$ be the external direct sum of $U$ and $W$. show that $V$ is isomorphic to $\hat{V}$ under the correspondence $v=u+w\leftrightarrow(u,w)$

I face some trouble to solve this problems help me for understanding.
thank you.

1 reply
We2592
May 21, 2025
Acridian9
May 22, 2025
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D1028 : A strange result about linear algebra
Dattier   2
N May 11, 2025 by ysharifi
Source: les dattes à Dattier
Let $p>3$ a prime number, with $H \subset M_p(\mathbb R), \dim(H)\geq 2$ and $H-\{0\} \subset GL_p(\mathbb R)$, $H$ vector space.

Is it true that $H-\{0\}$ is a group?
2 replies
Dattier
May 10, 2025
ysharifi
May 11, 2025
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Putnam 2000 B1
ahaanomegas   9
N Apr 25, 2025 by Ilikeminecraft
Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.
9 replies
ahaanomegas
Sep 6, 2011
Ilikeminecraft
Apr 25, 2025
J
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linear transformation
We2592   0
Apr 25, 2025
Q) let $V$ be one dimensional vector space over field $\mathbb{F}$ then find all linear mapps possible on
$T:V \to V $? generalize it for n dimensional ?

Q)let $T:\mathbb{R}^4 \to \mathbb{R}$ be given by $T(x_1,x_2,...,x_n)=x_i$ for a fixed i. then show that $T_i$ is a linear map? generalize it.

Q)let ${\{v_i}\}_{i=1}^{n}$ be a basis of $V$.Define $T_i:V \to \mathbb{R}$ by $T_i(v)=a_i$ if $v=a_1v_1+...+a_nv_n$ then show that $T_i$ is a linear map.

Q)let $f_i:\mathbb{R}^m \to \mathbb{R}$ be arbitary functions. let $T:\mathbb{R}^m \to \mathbb{R}^n$ be defined by $T(x_1,...,x_m)=(f_1(x),...,f_n(x))$ , when $T$ is linear?

how to solve help
0 replies
We2592
Apr 25, 2025
0 replies
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Basis and dimension
We2592   1
N Apr 14, 2025 by Etkan
Q) prove that $V:=\mathbb{R(\mathbb{Q})}$ is not a finite dimentional vector space.

Is we replace Q by $Q^{c}$ can it still be a vector space of finite dimensional?
1 reply
We2592
Apr 14, 2025
Etkan
Apr 14, 2025
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Putnam 2003 B1
btilm305   13
N Apr 13, 2025 by clarkculus
Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?
13 replies
btilm305
Jun 23, 2011
clarkculus
Apr 13, 2025
J
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vector space 2
We2592   3
N Apr 12, 2025 by Squeeze
1Q) let the vector subspace $W_1=\{(a_1,a_2,a_3)\in \mathbb{R}^3\mid 2a_1-7a_2+a_3=0\}$ of $\mathbb{R}^3$ then find the a subspace of $W_2$ of $\mathbb{R}^3$ such that $\mathbb{R}^3=W_1\times W_2$

2Q)Give two example of linearly independent set having more than one elementfor the vector space $P(\{1,2,3\})$ over $\mathbb{Z}_2$

3)find a subset $S$ of $\mathbb{R}$ which is L.D in the vector space $\mathbb{R}_\mathbb{R}$ but L.I. in the vector sapce of $\mathbb{R}_\mathbb{Q}$. And vise versa?

4)find two subspaces $X$ and $Y$ of $\mathbb{R}^3$ such that $\mathbb{R}^3=X+Y$ but $\mathbb{R}^3\neq X\times Y$

now what is the intuition should be in mind to solve this kind of problem or guessing or looking patterns?
3 replies
We2592
Apr 11, 2025
Squeeze
Apr 12, 2025
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vectorspace
We2592   1
N Apr 10, 2025 by Acridian9
Q.) Let $V = \{ (x, y) \mid x, y \in \mathbb{F} \}$
where $\mathbb{F}$ is field. Define addition of elements of $V$ coordinate wise and for $C\in\mathbb{F}$ and $x,y\in V$ define $c(x,y)=(x,0)$.

Is $V$ is a vector space over field $\mathbb{F}$

how to solve it please help
1 reply
We2592
Apr 10, 2025
Acridian9
Apr 10, 2025
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