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There is a theorem about the number of bases of a general matroid: Theorem: An $n$ element matroid with rank $r$, in which any $s$ element set is independent, has at least $\binom{n−r+s}{s}$ bases; e …

It seems to me that Robert Israel's argument generalizes. The sum of any $\frac{n}{2}$ vectors in your collection has to be distinct, in order for any $n$ to be linearly independent. From this one get …

S. Cater proved that every $\mathbb F$ valued map $f$ on square matrices which satisfies $f(ABC)=f(CBA)$ can be written as $f(X)=\pi(\det(X))$ for a unique map $\pi:\mathbb F\to \mathbb F$. Also $f$ i …

Tony already mentioned that the maximum size of a set of vectors that are $k$-wise linearly independent over a finite field $\mathbb F_q$ grows linearly with $q$. In our situation, however, this is …

For a matroid the elements that are contained in every basis are called coloops, dual to the notion of a loop, which is an element not contained in any basis. Since you are interested in linearly inde …

Here is a start, suppose that $V_i$ is $\mathbb C^{k_i}$ (and restricting to $k_1,k_2,\dots,k_n, n\geq 2$). The tuples $(k_1,k_2,\dots,k_n)$ for which the action of $GL_{k_1}\times\cdots\times GL_{k_n …

You already have the answer and proof, so let me just use this post to advertise some basic graph theory :). Take a cycle graph on $N$ vertices, and weigh each edge by $\Delta_i$, $i=1,2,\dots N$. The …

This is true, and in fact you can show a slightly more general fact: $$\det_{1\le i,j\le n}\left( \binom{x_i}{2j}+ \binom{-x_i}{2j}\right)=\prod_{i=1}^n x_i^2 \prod_{i<j} (x_j^2-x_i^2) \prod_{j=1}^n …

The diagonal entries of a positive definite matrix are real and non-negative. If we let the rows of $w$ be $v_1,\dots,v_n$, then the diagonal entries of $I_n-w\overline{w}$ are $1-v_i\overline{v_i}=1- …

In response to Qiaochu's question in the comments, Fong and Sourour prove in their paper The group generated by unipotent operators that every element of $\mathrm{SL}_n(\mathbb C)$ is a product of thr …

You are looking at the vector of coefficients of the polynomial $$\sum_{\epsilon}\prod_{i=1}^{m+2}(1-\epsilon_i x_i)$$ where $\epsilon$ runs over all choices of signs $\pm$ provided there are exactly …

To each such matrix $A$ we can associate $A_1,A_2$, the sets of column indices and row indices respectively. The families $\mathcal F_i=\{A_i| A\in \mathcal F\}$ are intersecting families of subsets, …

Let's denote by $S_{k,n}$ the set of possible integers $m$, such that $\mathbb R^k$ can be divided into $m$ regions by $n$ hyperplanes. If we denote by $S^{P}_{k,n}$ the set defined similarly but for …

Let's denote the vector $v=(1,1,\dots,1)$. Then we have that $$\det(A_n-av^{\intercal}v)=\prod_{k=1}^n(c_k-a) \quad \textrm{and}\quad\det(A_n-bv^{\intercal}v)=\prod_{k=1}^n(c_k-b)$$ since they are low …

I've seen your $D_k$ be called a subdiscriminant. Similarly the principal minors of the Sylvester matrix of two polynomials are the subresultants. The result you want is that subdiscriminants are (up …