document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1034158>Question 1034158</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that a line through the origin of R^3 is a subspace of R^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #648771 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=648771\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let L be any line in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > that passes through the origin.  Then it would have the symmetric equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fa+=+y%2Fb+=+z%2Fc ALIGN=MIDDLE  ALT=\"x%2Fa+=+y%2Fb+=+z%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <a,b,c> as its direction vector.<BR>\n" );
document.write( "Let w and v be two vectors in L.  Then w = k(a,b,c) and v = l(a,b,c) for some constants k and l.<BR>\n" );
document.write( "Now a non-empty subset of any vector space is a subspace iff <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Aw%2Bbeta%2Av ALIGN=MIDDLE  ALT=\"alpha%2Aw%2Bbeta%2Av\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also in the subset for any two vectors w and v in the said subset.\r<BR>\n" );
document.write( "\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Aw%2Bbeta%2Av+ ALIGN=MIDDLE  ALT=\"alpha%2Aw%2Bbeta%2Av+\"   DISABLED_style= \"max-width:90%; height:auto;\" >= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >*k(a,b,c) + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >*l(a,b,c)\r<BR>\n" );
document.write( "\n" );
document.write( "= (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >*k + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >*l)(a,b,c),\r<BR>\n" );
document.write( "\n" );
document.write( "meaning the resulting linear combination is still in the line L.\r<BR>\n" );
document.write( "\n" );
document.write( "Hence any line through the origin of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a subspace of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );