2.4c - 1 Math 110 Homework 6 Partial Solutions If you have...

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1 Math 110 Homework 6 Partial Solutions If you have any questions about these solutions, or about any problem not solved, please ask via email or in office hours, etc. 2.4.14 Define T : V F 3 by T ±± a a + b 0 c ²² = ( a,b,c ) It is easy to show that this is linear (I will omit the proof). I need only now show that T is one-to-one and onto. Suppose that ( x,y,z ) F 3 . Then notice that T ±± x y - x 0 z ²² = ( x,y,z ). Thus T is onto. Now compute N ( T ). If T ±± a a + b 0 c ²² = (0 , 0 , 0), then a = 0, a + b = 0 and c = 0. This means that ± a a + b 0 c ² = ± 0 0 0 0 ² and that T is one-to-one. Thus T is an isomorphism. 2.4.16 First we show that Φ is linear. Suppose that A,C M n × n ( F ). Then Φ( A + C ) = B - 1 ( A + C ) B = B - 1 AB + B - 1 CB = Φ( A ) + Φ( C ). Further, if c F , then Φ( cA ) = B - 1 ( cA ) B = cB - 1 AB = c Φ( A ). Since Φ is a linear transformation between two vector spaces with the
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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2.4c - 1 Math 110 Homework 6 Partial Solutions If you have...

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