m235notes5 - Notes 5: First Properties of Matrices and...

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Notes 5: First Properties of Matrices and their Geometry If two matrices have the same size, then we can add them by adding componentwise. Example 1 . Let A = 2 3 - 1 0 4 7 , B = - 1 - 3 - 2 - 4 - 4 - 5 , then A + B = 1 0 - 3 - 4 0 2 . We can multiply a matrix by a scalar. Example 2 . Let A = 2 3 - 1 0 4 7 R , then λA = 2 3 - 1 0 4 7 = λ 2 λ 3 λ ( - 1) 0 λ 4 λ 7 . If A is an n × m matrix, and u,v R m , then A ( u + v ) = Au + Av . Example 3 . Let A = 2 3 - 1 0 4 7 ,u = ± u 1 u 2 ² ,v = ± v 1 v 2 ² . We first compute Au + Av = 2 u 1 + 3 u 2 - 1 u 1 + 0 u 2 4 u 1 + 7 u 2 + 2 v 1 + 3 v 2 - 1 v 1 + 0 v 2 4 v 1 + 7 v 2 . Second we compute A ( u + v ) = 2 3 - 1 0 4 7 ± u 1 + v 1 u 2 + v 2 ² = 2( u 1 + v 1 ) + 3( u 2 + v 2 ) ( - 1)( u 1 + v 1 ) + 0( u 2 + v 2 ) 4( u 1 + v 1 ) + 7( u 2 + v 2 ) . These are the same because of the distributive law This example is not a proof. We omit giving a proof. If A is an n × m matrix, u R m R , then A ( λu ) = λA ( u ) 1
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Example 4 . Again let A = 2 3 - 1 0 4 7 , let λ R , and let u = ± u 1 u 2 ² R 2 . We have A ( λu ) = A ± λu 1 λu 2 ² = 2 λu 1 + 3 λu 2 ( - 1) λu 1 + 0 λu 2 4 λu 1 + 7 λu 2 on the one hand, and on the other hand we have λA ( u ) = λ 2 3 - 1 0 4 7 ± u 1 u 2 ² = λ 2 u 1 + 3 u 2 - 1 u 1 + 0 u 2 4 u 1 + 7 u 2 = λ (2 u 1 + 3 u 2 ) λ ( - 1 u 1 + 0 u 2 ) λ (4 u 1 + 7 u 2 ) .
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m235notes5 - Notes 5: First Properties of Matrices and...

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