document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167649>Question 1167649</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose u, v ∈ R3. Determine if the function<BR>\n" );
document.write( "<<u,v>> = 2u1v1 + u2v2 + 4u3v3<BR>\n" );
document.write( "is an inner product on R3. If it is not an inner product, list the axioms which do not hold. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852211 by </B><A HREF=/tutors/aboutme.mpl?userid=Resolver123><B>Resolver123(6)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Resolver123><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852211\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are given a function defined on <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;\" > as:\r<BR>\n" );
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document.write( "(u, v) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5B1%5Dv%5B1%5D+%2B+u%5B2%5Dv%5B2%5D+%2B+4u%5B3%5Dv%5B3%5D ALIGN=MIDDLE  ALT=\"2u%5B1%5Dv%5B1%5D+%2B+u%5B2%5Dv%5B2%5D+%2B+4u%5B3%5Dv%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We want to determine whether this function defines an inner product on <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;\" >. Recall the inner product axioms.\r<BR>\n" );
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document.write( "A function (.,.) : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3+%2AR%5E3 ALIGN=MIDDLE  ALT=\"R%5E3+%2AR%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an inner product if it satisfies the following axioms for all u, v, w 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;\" > and all scalars <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "1. Symmetry: (u, v) = (v, u)\r<BR>\n" );
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document.write( "2. Linearity in the first argument (a.k.a. \"bilinearity\" for real vector spaces): (c*u + w, v) = c*(u, v) + (w, v)\r<BR>\n" );
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document.write( "3. Positive-definiteness: (u, u)  ≥ 0 and (u,u)=0 if and only if u = 0.\r<BR>\n" );
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document.write( "1.) Check Symmetry\r<BR>\n" );
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document.write( "Compute both sides:\r<BR>\n" );
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document.write( "(u, v) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5B1%5Dv%5B1%5D+%2B+u%5B2%5Dv%5B2%5D+%2B+4u%5B3%5Dv%5B3%5D%0D%0A=+2v%5B1%5Du%5B1%5D+%2B+v%5B2%5Du%5B2%5D+%2B+4v%5B3%5Du%5B3%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = (v,u).\r<BR>\n" );
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document.write( "Therefore, symmetry holds.\r<BR>\n" );
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document.write( "2.) Check Linearity in First Argument\r<BR>\n" );
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document.write( "Let u, w, v be 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;\" > and c be in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let’s compute:\r<BR>\n" );
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document.write( "(c*u + w, v ) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28c%2Au%5B1%5D+%2B+w%5B1%5D%29v%5B1%5D+%2B+%28c%2Au%5B2%5D+%2B+w%5B2%5D%29v%5B2%5D+%2B+4%28c%2Au%5B3%5D+%2B+w%5B3%5D%29v%5B3%5D=%28c%2A2%2Au%5B1%5D%2Av%5B1%5D+%2Bc%2Au%5B2%5D%2Av%5B2%5D+%2Bc%2A4%2Au%5B3%5D%2Av%5B3%5D%29+%2B+%282%2Aw%5B1%5D%2Av%5B1%5D%2Bw%5B2%5D%2Av%5B2%5D%2B4%2Aw%5B3%5D%2Av%5B3%5D%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = c*(u,v) + (w,v).\r<BR>\n" );
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document.write( "Therefore, linearity in the first argument holds.\r<BR>\n" );
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document.write( "3.) Check for Positive-Definiteness\r<BR>\n" );
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document.write( "It must be shown that:\r<BR>\n" );
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document.write( "(u, u) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5B1%5D%5E2+%2B+u%5B2%5D%5E2+%2B+4u%5B3%5D%5E2+ ALIGN=MIDDLE  ALT=\"2u%5B1%5D%5E2+%2B+u%5B2%5D%5E2+%2B+4u%5B3%5D%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ≥ 0 and  = 0 iff u = 0.\r<BR>\n" );
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document.write( "Note that each term is squared and multiplied by a positive scalar, so the whole expression is non-negative.\r<BR>\n" );
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document.write( "ALso, if we let (u,u) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5B1%5D%5E2+%2B+u%5B2%5D%5E2+%2B+4u%5B3%5D%5E2=0+ ALIGN=MIDDLE  ALT=\"2u%5B1%5D%5E2+%2B+u%5B2%5D%5E2+%2B+4u%5B3%5D%5E2=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then this statement is true if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B1%5D=u%5B2%5D+=+u%5B3%5D+=+0 ALIGN=MIDDLE  ALT=\"u%5B1%5D=u%5B2%5D+=+u%5B3%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, i.e., u = (0,0,0). \r<BR>\n" );
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document.write( "Therefore, positive-definiteness holds.\r<BR>\n" );
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document.write( "Since all three axioms (symmetry, linearity, positive-definiteness) are satisfied, (u,v) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5B1%5Dv%5B1%5D+%2B+u%5B2%5Dv%5B2%5D+%2B+4u%5B3%5Dv%5B3%5D ALIGN=MIDDLE  ALT=\"2u%5B1%5Dv%5B1%5D+%2B+u%5B2%5Dv%5B2%5D+%2B+4u%5B3%5Dv%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an inner product on <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" );
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