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<font size="3">Considerem el subespai de les matrius </font><font size="3">simètriques d'ordre 2</font><font size="3"> amb coeficients reals.<br /><br />Una base d'aquest espai és <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfenced close=¨}¨ open=¨{¨»«mrow»«mfenced»«mtable»«mtr»«mtd»«mn»1«/mn»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»,«/mo»«mfenced»«mtable»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»1«/mn»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»,«/mo»«mfenced»«mtable»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mfenced»«/math»</span></span><span class="nolink"><span class="nolink"></span></span>.<br /><br />a) Quina és la dimensió d'aquest espai?<br /><br />b) Considerem ara la matriu <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mtable»«mtr»«mtd»«mi»#a«/mi»«/mtd»«mtd»«mi»#b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»#b«/mi»«/mtd»«mtd»«mi»#c«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/math»</span></span>. Determineu la seva expressió en la base <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»1«/mn»«/msub»«/math»</span></span>.<br /></font> <div style="text-align: center"> <div style="text-align: justify"><font size="3"><span class="nolink"></span></font><br /><font size="3">c) Denotem per <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»u«/mi»«mi»i«/mi»«/msub»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«mo»,«/mo»«mn»3«/mn»«/math»</span></span>, els elements de la base anterior i considerem la base donada per <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»v«/mi»«mi»j«/mi»«/msub»«mo»=«/mo»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mi»j«/mi»«/munderover»«msub»«mi»u«/mi»«mi»i«/mi»«/msub»«mo»§nbsp;«/mo»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»j«/mi»«mo»=«/mo»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«mo»,«/mo»«mn»3«/mn»«/math»</span></span> : <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mo»{«/mo»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«mo»,«/mo»«msub»«mi»v«/mi»«mn»2«/mn»«/msub»«mo»,«/mo»«msub»«mi»v«/mi»«mn»3«/mn»«/msub»«mo»}«/mo»«/math»</span></span>.<br />Calculeu les components de la matriu donada en l'apartat anterior respecte a aquesta nova base <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»2«/mn»«/msub»«/math»</span></span>.<br /><br />d) Donada la matriu que té per components {#aa,,#bb,#cc} respecte la base <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»2«/mn»«/msub»«/math»</span></span>, calculeu les components d'aquesta matriu respecte la base <span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»B«/mi»«mn»1«/mn»«/msub»«/math»</span></span>.<br /><br /><br /><span style="text-decoration: underline">Nota:</span> </font><font size="3">escriviu la solució entre claus tot separant cada apartat per una coma, és a dir, si la solució d'a) és 0, la solució de b) és (1,2,3), la de c) és (4,5,6) i la de d) és (7,8,9), hauríeu d'introduir com a solució<br /></font> <div style="text-align: center"><font size="3">{0,[1,2,3],[4,5,6],[7,8,9]}.</font><br /></div><br /></div><font size="3"><br /></font> <div style="text-align: justify"><font size="3"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></font><br /></div><font size="3"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></font></div> <div style="text-align: center"><font size="3"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></font><span class="nolink"><span class="nolink"><br /></span></span></div><br />
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66.66667%
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33.33333%
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20%
16.66667%
14.28571%
12.5%
11.11111%
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