document.write( "Question 1062030: Let P be a projection, so $\"P%5E2+=+P\"$ . If $\"c%3C%3E1\"$ , compute $\"%28I-cP%29%5E%28-1%29\"$ . \n" ); document.write( "

Algebra.Com's Answer #676786 by ikleyn(52817) $\"\"$ $\"About$

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\n" ); document.write( "Let P be a projection, so $\"P%5E2+=+P\"$ . If $\"c%3C%3E1\"$ , compute $\"%28I-cP%29%5E%28-1%29\"$ .
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\n" ); document.write( "\n" ); document.write( "I can do it for $\"abs%28c%29\"$ < 1. Again, let us assume that |c| < 1.\r
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document.write( "Then   =  = (since ) =  = \r\n" );
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document.write( "      ( apply the formula for the sum of an infinite geometric progression ) \r\n" );
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document.write( "      =  =  = \r\n" );
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document.write( "      =  =  = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. $\"%28I-cP%29%5E%28-1%29\"$ = $\"%281%2F%281-c%29%29%2AP\"$ .\r
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\n" ); document.write( "\n" ); document.write( "You may check it by making direct calculations that $\"%28I+-+cP%29\"$ * $\"%281%2F%281-c%29%29%2AP\"$ = I under the condition $\"P%5E2\"$ = $\"P\"$ .\r
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\n" ); document.write( "\n" ); document.write( "And after checking it directly (as I said) you may to extend the formula for |c| > 1. It is valid for |c| > 1, too.\r
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