query: Closed formula for $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)$
positive_passages.docid: 23071
positive_passages.text: Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
negative_passages.docid: 153612
negative_passages.text: Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$
negative_passages.docid: 141762
negative_passages.text: The direct sum of two closed subspace is closed? (Hilbert space)
negative_passages.docid: 29524
negative_passages.text: Formula for the sum $\sum_{i=2}^{n} \frac1{i^2-1}$
negative_passages.docid: 22933
negative_passages.text: Closed-form solution for the determinant of a Vandermonde-like matrix
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 562
negative_passages.text: Formula for calculating $\sum_{n=0}^{m}nr^n$
negative_passages.docid: 35564
negative_passages.text: Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$
negative_passages.docid: 155479
negative_passages.text: Summation of inner products
negative_passages.docid: 69980
negative_passages.text: Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
negative_passages.docid: 114575
negative_passages.text: If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
negative_passages.docid: 55659
negative_passages.text: Sum of closed and compact set in a TVS
negative_passages.docid: 2271
negative_passages.text: Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
negative_passages.docid: 67454
negative_passages.text: Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^n (x_i - \bar{x})(y_i)$.
negative_passages.docid: 19914
negative_passages.text: Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$