query: Why is $\dim_{\mathbb{Q}}\mathbb{R}^n=\operatorname{card}\mathbb{R}$ for all $n\geq1$?
positive_passages.docid: 26497
positive_passages.text: Take $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a basis for $\mathbb{R}$, then $|B|=2^{\aleph_0}$
negative_passages.docid: 54496
negative_passages.text: Showing that $\mathbb{R}$ and $\mathbb{R}\backslash\mathbb{Q}$ are equinumerous using Cantor-Bernstein
negative_passages.docid: 141258
negative_passages.text: Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?
negative_passages.docid: 25795
negative_passages.text: Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$
negative_passages.docid: 127198
negative_passages.text: What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?
negative_passages.docid: 141799
negative_passages.text: Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?
negative_passages.docid: 146847
negative_passages.text: Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
negative_passages.docid: 92740
negative_passages.text: prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
negative_passages.docid: 138256
negative_passages.text: What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?
negative_passages.docid: 128760
negative_passages.text: If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
negative_passages.docid: 104258
negative_passages.text: Proof Verification: $\dim\mathrm{Im} T+\dim\ker T=\dim V$
negative_passages.docid: 66272
negative_passages.text: Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?
negative_passages.docid: 66866
negative_passages.text: Algebraic dual of $\mathbb{R}^{\mathbb{N}}$
negative_passages.docid: 22149
negative_passages.text: How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
negative_passages.docid: 132653
negative_passages.text: How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
negative_passages.docid: 105014
negative_passages.text: Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?
negative_passages.docid: 45469
negative_passages.text: If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$