query: Difference between $\mathbb R^m\times\mathbb R^n$ and $\mathbb R^m\oplus\mathbb R^n$
positive_passages.docid: 18023
positive_passages.text: The direct sum $\oplus$ versus the cartesian product $\times$
negative_passages.docid: 6161
negative_passages.text: The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
negative_passages.docid: 109710
negative_passages.text: Bijection between $\mathbb N \times \mathbb N$ and $\mathbb N$
negative_passages.docid: 118288
negative_passages.text: Complex structure on $\mathbb S^1 \times \mathbb S^3$
negative_passages.docid: 70861
negative_passages.text: $\mathbb P^n\times \mathbb P^m\not\cong \mathbb P^{m+n}$
negative_passages.docid: 103308
negative_passages.text: Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$
negative_passages.docid: 4156
negative_passages.text: What is $\Bbb R^{\times}$? [unit group, ring to "times" power]
negative_passages.docid: 154754
negative_passages.text: Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?
negative_passages.docid: 31391
negative_passages.text: bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$
negative_passages.docid: 34768
negative_passages.text: Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$
negative_passages.docid: 154628
negative_passages.text: What should we understand from the definition of orthogonality in inner product spaces other than $\mathbb R^n$?
negative_passages.docid: 133820
negative_passages.text: The product topology on $\mathbb R_d\times \mathbb R$ vs. the dictionary order topology on $\mathbb R\times \mathbb R$
negative_passages.docid: 148685
negative_passages.text: Equivalences of definitions of submanifolds in $\mathbb{R^n}$
negative_passages.docid: 49166
negative_passages.text: $R^n \cong R^m$ iff $n=m$
negative_passages.docid: 132311
negative_passages.text: T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
negative_passages.docid: 21807
negative_passages.text: on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$
negative_passages.docid: 62303
negative_passages.text: For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$