37 - Matrices January 17, 2012 1 Motivation What is the...

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Unformatted text preview: Matrices January 17, 2012 1 Motivation What is the least number of required vectors to describe a point in R 3 ? What properties must the least number of vectors satisfy? Consider the equations, A 11 x 1 + A 12 x 2 + A 13 x 3 = y 1 , A 21 x 1 + A 22 x 2 + A 23 x 3 = y 2 , and A 31 x 1 + A 32 x 2 + A 33 x 3 = y 3 . If y 1 , y 2 and y 3 are given when can we solve for x 1 , x 2 and x 3 ? Definition 1.1. Given vector spaces V and W , the mapping L : V W is called Linear if and only if L ( a + b ) = L ( a ) + L ( b ) Example 1.1. Given a = ( a 1 ,a 2 ,a 3 ) R 3 , the mapping f : R 3 R defined by f ( b ) = a b = 3 i =1 a i b i is linear because a ( b + c ) = a b + a c and ( a b ) = ( a b ) . Definition 1.2. A real m m matrix A is a rectangular array with m rows and n columns whose entries are real numbers. A = A 11 A 12 ... A 1 n A 21 A 22 ... A 2 n . ....
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37 - Matrices January 17, 2012 1 Motivation What is the...

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