Intereting Posts

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?
$m$ balls $n$ boxes probability problem
Derivative of the $f(x,y)=\min(x,y)$
Matrix P to the power of 4, i.e $P^4$, is this the same as $P^2 \cdot P^2$?
Application of maximum modulus principle
Finding out this combination
number of non-isomorphic rings of order $135$
Are continuous functions with compact support bounded?
How to evaluate $\int_{0}^{1} \frac{\ln x}{x+1} dx$
wildly ramified extensions and $p$-power roots of unity
Find $\lim_{x \to 0} x \cdot \sin{\frac{1}{x}}\cos{\frac{1}{x}}$
Why is it hard to prove whether $\pi+e$ is an irrational number?
Find prime ideals
Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)
What is the integral of $\int e^x\,\sin x\,\,dx$?

Let $n$ be a positive integer, and $A$ be the set of all non-zero vectors of the form $(e_1,e_2,\dots,e_n)$, where $ e_i\in\{0,1\}$. So $|A|=2^n-1$. Let $B$ be a proper subset of $A$. Does there always exists a subset $C\subseteq A$ of size at most $n-1$ such that for any vector in $b\in B$, there exists a vector $c\in C$ with $c\cdot b=1$?

If we were to allow $B=A$, then we can choose a set $C$ of size $n$ to be the unit vectors $(1,0,\dots,0),(0,1,\dots,0),\dots,(0,0,\dots,1)$. Now, since we have $B$ is a proper subset of $A$, it seems we should also need a smaller $C$.

- Evaluating 'combinatorial' sum
- Definition of the Young Symmetrizer
- Estimation of a combinatorial sum
- Bijective proof for equal number of odd-sized/even-sized subsets of a finite set
- Permutation isomorphic subgroups of $S_n$ are conjugate
- Number of simple paths between two vertices on an $n \times m$ square-grid graph?

- Product of simplicial complexes?
- Logical formula of definition of linearly dependent
- Counterexamples for “every linear map on an infinite dimensional complex vector space has an eigenvalue”
- Sum of the series $\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+…$
- What are mandatory conditions for a family of matrices to commute?
- How to solve second degree recurrence relation?
- What is the rank of COCHIN
- Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,B_1,…,A_k,B_k)$ for some $k$ when $S$ is symplectic
- Limit of matrix powers.
- How can you explain the Singular Value Decomposition to Non-specialists?

Sure. We basically use a missing vector to define things in a way that saves us 1 member of $C$.

More formally, let $u$ be a vector missing from $B$, $S$ be the support of $u$, and let $v_i$ be the vector which has only one $1$ in the $i$-th position. Then define $C$ as follows:

- include in $C$ all vectors $v_i$ for $i\notin S$;
- fix some $s\in S$, and include in $C$ all vectors of the form $v_s + v_j$ where $j\in S-\{s\}$.

This gives a total of $|C|=n-1$ elements. Now let $v\in B$ be any vector. If $v_i=1$ for some $i\notin S$, we can just use $v_i\in C$. Otherwise, the support of $v$ is a subset of $S$. If $v_s\neq 1$, there will be some $j$ such that $v_j=1$ and we can use $v_s+v_j\in C$.

Otherwise, $v_s=1$; **this is the key step:** this means there will be $j\in S$ such that $v_j=0$, since otherwise $v=u$. Thus, we can use $v_s+v_j\in C$.

This covers all cases and completes the analysis.

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- Question on an isomorphism in the proof that $k \cong k \otimes_k k$
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- Classic Hand shake question
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- An integral of a rational function of logarithm and nonlinear arguments
- Regulators and uniqueness
- Is there any mathematical meaning in this set-theoretical joke?
- Representing IF … THEN … ELSE … in math notation
- Bezout's Theorem
- If $ad$ and $bc$ are odd and even, respectively, then prove that $ax^3+bx^2+cx+d$ has an irrational root.