Engineering Calculus Notes 239

Engineering Calculus Notes 239 - i th entry oF the row with...

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3.2. LINEAR AND AFFINE FUNCTIONS 227 Matrix Representation of Linear Functions We are now going to set up what will at frst look like an unnecessary complication oF the picture above, but in time it will open the door to appropriate generalizations. The essential data concerning a linear Function ( a.k.a. a homogeneous polynomial oF degree one) is the set oF values taken by on the standard basis oF R 3 : a i = ( −→ e i ) , i = 1 , 2 , 3 . We shall Form these numbers into a 1 × 3 matrix (a row matrix ), called the matrix representative oF : [ ] = b a 1 a 2 a 3 B . We shall also create a 3 × 1 matrix (a column matrix ) whose entries are the components oF the vector −→ x , called the coordinate matrix oF −→ x : [ −→ x ] = x 1 x 2 x 3 . We then defne the product oF a row with a column as the result oF substituting the entries oF the column into the homogeneous polynomial whose coe±cients are the entries oF the row; equivalently, we match the
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Unformatted text preview: i th entry oF the row with the i th entry oF the column, multiply each matched pair, and add: b a 1 a 2 a 3 B x 1 x 2 x 3 = a 1 x 1 + a 2 x 2 + a 3 x 3 . OF course, in this language, we are representing the linear Function ℓ : R 3 → R as the product oF its matrix representative with the coordinate matrix oF the input ℓ ( −→ x ) = [ ℓ ][ −→ x ] . Another way to think oF this representation is to associate, to any row , a vector −→ a (just put commas between the entries oF the row matrix), and then to notice that the product oF the row with the coordinate matrix oF −→ x is the same as the dot product oF the vector −→ a with −→ x : b a 1 a 2 a 3 B x 1 x 2 x 3 = ( a 1 ,a 2 ,a 3 ) · ( x 1 ,x 2 ,x 3 ) = −→ a · −→ x ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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