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Unformatted text preview: vector space (see chapter 4). 1.2. Some multiplication properties. Some more properties involving multiplication: Theorem 2. • ( AB ) C = A ( BC ) • ( A + B ) C = AC + BC and A ( B + C ) = AB + AC 1.3. Things which fail. However, here are some things which are FALSE! : • It is not true that AB = BA . For example, ± 1 0 0 0 ² · ± 0 1 0 0 ² 6 = ± 0 1 0 0 ² · ± 1 0 0 0 ² • AB = 0 doesn’t mean that either A = 0 or B = 0, for example ± 0 1 0 0 ² · ± 1 0 0 0 ² = ± 0 0 0 0 ² • AB = AC might be true even if B 6 = C and A 6 = 0. For instance ± 1 0 0 0 ² · ± 1 1 1 1 ² = ± 1 0 0 0 ² · ± 1 1 0 0 ² but ± 1 1 1 1 ² 6 = ± 1 1 0 0 ² 1 (so you can’t just “cancel” the matrix) ± 1 0 0 0 ² 1.4. Algbraic properties of the transpose. Theorem 3. • ( A T ) T = A • ( A + B ) T = A T + B T • ( AB ) T = B T A T NOTE THIS ONE CAREFULLY!!! • ( rA ) T = rA T 2...
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This note was uploaded on 03/31/2012 for the course MA 265 taught by Professor Bens during the Spring '08 term at Purdue.
 Spring '08
 Bens
 Linear Algebra, Algebra, Matrices

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