5.1

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B>Álgebra lineal</B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="2" COLOR="#FFFFCC" FACE="Bell MT" ><B>   </B></FONT><B> |     </B><A HREF="index.htm" TARGET="_top" TITLE="Á l g e b r a l i n e a l"><U><B>Página principa</B></U></A><B>l  |  </B><A HREF="id17.htm" TARGET="_top" TITLE="Contenido"><U><B>Contenido</B></U></A><B>  |  </B><A HREF="id26.htm" TARGET="_top" TITLE="Stanley Grossman"><U><B>Grossman</B></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="id28.htm" TARGET="_top" TITLE="Problemas 1.1"><U><B>Problemas 1.1</B></U></A><B>   |   </B><A HREF="id29.htm" TARGET="_top" TITLE="1.2"><U><B>1.2</B></U></A><B>   |   </B><A HREF="id30.htm" TARGET="_top" TITLE="1.4"><U><B>1.4</B></U></A><B>   |   </B><A HREF="id32.htm" TARGET="_top" TITLE="2.1"><U><B>2.1</B></U></A><B>   |   </B><B>5.1</B></div> <div> </div> </td> </tr><tr> <td valign="top" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="5543838917"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> </td> <td valign="top" height=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Problemas 5.1</B></div> <div> <B>Espacios vectoriales</B></div> <div> <B>Propiedades y definiciones básicas</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="9294650114"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>Apuntes básicos</B></div> <bR> <div> <FONT SIZE="4" COLOR="#009933" FACE="Bell MT" ><B>Definición 5.1.1.</B></FONT><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B> Espacio vectorial real.  </B></FONT>Un <I>espacio vectorial real </I><I>V</I><I> </I>es un conjunto de objetos, denominados <I>vectores</I>, junto con dos operaciones binarias llamadas <I>suma</I> y <I>multiplicación por un escalar</I>, y que satisfacen los diez axiomas siguientes:</div> <div> <B>Axiomas de un espacio vectorial</B>.</div> <div> i)     Cerradura bajo la suma</div> <div> <TABLE BORDER="1" CELLPADDING="0" CELLSPACING="1" WIDTH="346" BORDERCOLORLIGHT="#003366" BORDERCOLORDARK="#003366" FRAME="BOX" RULES="ALL" ALIGN="RIGHT" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 287 ><NOBR><div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="1163033713"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> </NOBR></tD> </tR> </TABLE></div> <bR> <div> <IMG SRC="4bd1d170.gif" border=0 width="285" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> ii)     Ley asociativa de la suma de vectores</div> <div> <IMG SRC="4be871c0.gif" border=0 width="391" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> iii)    El <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" ><B>0 </B></FONT>se llama vector cero o idéntico aditivo </div> <div> <IMG SRC="4bfe8170.gif" border=0 width="488" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> iv)    -<B>x</B> se llama inverso aditivo de <B>x</B></div> <div> <B><IMG SRC="4c0ac1c0.gif" border=0 width="428" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> v)     Ley conmutativa de la suma de vectores</div> <div> <B><IMG SRC="4c24d170.gif" border=0 width="333" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> vi)    Cerradura bajo la multiplicación por un escalar</div> <div> <IMG SRC="4c359170.gif" border=0 width="345" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> vii)   Primera ley distributiva</div> <div> <IMG SRC="4c4101c0.gif" border=0 width="528" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> viii)  Segunda ley distributiva</div> <div> <IMG SRC="4c5d81c0.gif" border=0 width="472" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> ix)    Ley asociativa de la multiplcación por escalares</div> <div> <IMG SRC="4c8351c0.gif" border=0 width="565" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="4ca11170.gif" border=0 width="273" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>Enunciados de los problemas 5.1</B></div> <bR> <div> <B>Lista de los Problemas del álgebra lineal (Ed. 7) de Stanley Grossman</B></div> <div> <B>|</B> <A HREF="id28.htm" TARGET="_top" TITLE="Problemas 1.1"><U>Problemas 1.1</U></A> | <A HREF="id29.htm" TARGET="_top" TITLE="1.2"><U>Problemas 1.2</U></A> | <A HREF="id30.htm" TARGET="_top" TITLE="1.4"><U>Problemas 1.4</U></A> | <A HREF="id32.htm" TARGET="_top" TITLE="2.1"><U>Problemas 2.1</U></A> |  <A HREF="id112.htm" TARGET="_top" TITLE="5.1"><U>Problemas 5.1</U></A> |</div> <bR> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="0f028280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="0f128280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="http://www.youtube.com/user/MatematicaAp"><U><IMG SRC="0f228280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <a href="HTMLobj-1206/Aviso_Baldor.docx"><img align="bottom" border="0" width="40" height="40" src="HTMLobj-1206/gato_1.jpg"></a>  </div> <div>  <a href="http://twitter.com/Calculo21" class="twitter-follow-button" data-button="grey" data-text-color="#FFFFFF" data-link-color="#00AEFF" data-show-count="false" data-lang="es">Follow @Calculo21</a> <script src="http://platform.twitter.com/widgets.js" type="text/javascript"></script>  </div> <bR> <div> <B>|  </B><A HREF="index.htm" TARGET="_top" TITLE="Á l g e b r a l i n e a l"><U><B>Página principal</B></U></A><B>  |  </B><A HREF="id17.htm" TARGET="_top" TITLE="Contenido"><U><B>Contenido</B></U></A><B>   |   </B><A HREF="id18.htm" TARGET="_top" TITLE="email"><U><B>e-mail</B></U></A><B>   |   </B><A HREF="id19.htm" TARGET="_top" TITLE="Enlaces"><U><B>Enlaces</B></U></A><B>  |</B></div> <bR> </td> </tr></table></body>