http://www.exampleproblems.com/wiki/index.php?title=Alg7.1&feed=atom&action=history Alg7.1 - Revision history 2022-05-20T08:17:09Z Revision history for this page on the wiki MediaWiki 1.33.0 http://www.exampleproblems.com/wiki/index.php?title=Alg7.1&diff=1669&oldid=prev Todd at 23:25, 24 October 2005 2005-10-24T23:25:23Z <p></p> <p><b>New page</b></p><div>====Step 1====<br /> Note that &lt;math&gt;\mathbb{R}&lt;/math&gt; is a vector space over the field &lt;math&gt;\mathbb{Q}&lt;/math&gt;<br /> of rational numbers, with ordinary real multiplication as the scalar multiplication. Let ''H'' be the basis of that space (a so-called ''Hamel basis'').<br /> The dimension over &lt;math&gt;\mathbb{Q}&lt;/math&gt; of the defined space is &lt;math&gt;|H| = c = 2^{\aleph_0}&lt;/math&gt;.<br /> <br /> ====Step 2====<br /> Note that &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is also a vector space over the field &lt;math&gt;\mathbb{Q}&lt;/math&gt;; it has a basis &lt;math&gt;(H\times\{0\}) \cup (\{0\}\times H)&lt;/math&gt;<br /> so its dimension is &lt;math&gt;|(H\times\{0\}) \cup (\{0\}\times H)| = 2|H| = |H|&lt;/math&gt; because &lt;math&gt;H&lt;/math&gt; is infinite, so this dimension is &lt;math&gt;c&lt;/math&gt; too.<br /> <br /> ====Step 3====<br /> Both &lt;math&gt;\mathbb{R}&lt;/math&gt; and &lt;math&gt;\mathbb{R}^2&lt;/math&gt; are vector spaces over <br /> &lt;math&gt;\mathbb{Q}&lt;/math&gt; of the same dimension; therefore they are isomorphic (as vector spaces). However, every isomorphism of linear spaces is additive and for that reason it is<br /> also an isomorphism of their additive groups. We have proven that &lt;math&gt;(\mathbb{R},+) \simeq (\mathbb{R}^2,+)&lt;/math&gt;.<br /> <br /> ====Step 4====<br /> The isomorphism between &lt;math&gt;(\mathbb{R}^n,+)&lt;/math&gt; and &lt;math&gt;(\mathbb{R}^m,+)&lt;/math&gt; is a trivial consequence of the isomorphism proven above.<br /> <br /> [[Abstract Algebra]]&lt;br&gt;&lt;br&gt;<br /> <br /> [[Main Page]]</div> Todd