document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167650>Question 1167650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A=<BR>\n" );
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document.write( "| -1  3 | </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852198 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=852198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have to show that the inner product defined by ⟨𝑢,𝑣⟩ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28Au%29%5ET%2A%28Av%29 ALIGN=MIDDLE  ALT=\"%28Au%29%5ET%2A%28Av%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > produces the expression:\r<BR>\n" );
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document.write( "⟨𝑢,𝑣⟩ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5u%5B1%5Dv%5B1%5D-u%5B1%5Dv%5B2%5D-u%5B2%5Dv%5B1%5D%2B10u%5B2%5Dv%5B2%5D ALIGN=MIDDLE  ALT=\"5u%5B1%5Dv%5B1%5D-u%5B1%5Dv%5B2%5D-u%5B2%5Dv%5B1%5D%2B10u%5B2%5Dv%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%28matrix%282%2C1%2Cu%5B1%5D%2Cu%5B2%5D%29%29 ALIGN=MIDDLE  ALT=\"u+=+%28matrix%282%2C1%2Cu%5B1%5D%2Cu%5B2%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+%28matrix%282%2C1%2Cv%5B1%5D%2Cv%5B2%5D%29%29 ALIGN=MIDDLE  ALT=\"v+=+%28matrix%282%2C1%2Cv%5B1%5D%2Cv%5B2%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "These imply that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Au+=+%28matrix%282%2C2%2C2%2C1%2C-1%2C3%29%29%2A%28matrix%282%2C1%2Cu%5B1%5D%2Cu%5B2%5D%29%29=%28matrix%282%2C1%2C2u%5B1%5D%2Bu%5B2%5D%2C-u%5B1%5D%2B3u%5B2%5D%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Av+=+%28matrix%282%2C2%2C2%2C1%2C-1%2C3%29%29%2A%28matrix%282%2C1%2Cv%5B1%5D%2Cv%5B2%5D%29%29=%28matrix%282%2C1%2C2v%5B1%5D%2Bv%5B2%5D%2C-v%5B1%5D%2B3v%5B2%5D%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "These give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28Au%29%5ET%2A%28Av%29=%28matrix%281%2C2%2C2u%5B1%5D%2Bu%5B2%5D%2C-u%5B1%5D%2B3u%5B2%5D%29%29%2A%28matrix%282%2C1%2C2v%5B1%5D%2Bv%5B2%5D%2C-v%5B1%5D%2B3v%5B2%5D%29%29=%28+2u%5B1%5D%2Bu%5B2%5D%29%2A%282v%5B1%5D%2Bv%5B2%5D%29%2B%28+-u%5B1%5D%2B3u%5B2%5D%29%2A%28-v%5B1%5D%2B3v%5B2%5D%29+=+5u%5B1%5Dv%5B1%5D-u%5B1%5Dv%5B2%5D-u%5B2%5Dv%5B1%5D%2B10u%5B2%5Dv%5B2%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The proof is complete. </SPAN>\n" );
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