query: Prove that $(x_1, y_1)$ and $(x_2, y_2)$ form a basis of $\mathbb Z \oplus \mathbb Z$ iff $x_1y_2 - y_2x_1 = \pm 1$
positive_passages.docid: 148101
positive_passages.text: Basis for $\mathbb{Z}^2$
negative_passages.docid: 158852
negative_passages.text: If $\{v_1, v_2, v_3\}$ is an orthonormal basis of $\mathbb{R}^3$ and $v = a_1v_1 + a_2v_2 + a_3v_3$, can we have $|a_1| > 3$?
negative_passages.docid: 122581
negative_passages.text: Prove or disprove that $G_1/H_1 \cong G_2/H_2$
negative_passages.docid: 53669
negative_passages.text: Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$
negative_passages.docid: 132487
negative_passages.text: Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
negative_passages.docid: 76188
negative_passages.text: Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$
negative_passages.docid: 157924
negative_passages.text: $\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
negative_passages.docid: 70475
negative_passages.text: Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.
negative_passages.docid: 117688
negative_passages.text: Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
negative_passages.docid: 62077
negative_passages.text: Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
negative_passages.docid: 121280
negative_passages.text: Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$
negative_passages.docid: 78873
negative_passages.text: How to show that the monomials are not a Schauder basis for $C[0,1]$
negative_passages.docid: 151805
negative_passages.text: Prove that $\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$
negative_passages.docid: 13421
negative_passages.text: Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$
negative_passages.docid: 8589
negative_passages.text: $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ not an integer
negative_passages.docid: 118299
negative_passages.text: Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
negative_passages.docid: 158156
negative_passages.text: Finding the dual basis for the space of polynomials of degree $\leq$ 2