207 - w = f (u) w = f (v) Example: with Example: with where...

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1 w = f ( u ) w = f ( v ) Example: with Example: with where
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2 Example: with T A ( u ) is the projection of u R 3 on to the xy -plane, which is the range of T A . Example: with shear transformation k = 2 Proof By the properties of the matrix-vector multiplication. Two special linear transformations: (1) the identity transformation I : R n R n with I ( x ) = x x R n . (2) the zero transformation T 0 : R n R m with T 0 ( x ) = 0 x R n . 7
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3 There are functions from R n to R m satisfying condition (ii) of the definition of linear transformation but not (i). For example, 2 22 for 0 otherwise x xy x T y xy y ⎡⎤ ⎛⎞ ⎢⎥ = + ⎜⎟ ⎣⎦ ⎝⎠ 0 If a function T : R n R m satisfies condition (i), then for all integers k , l > 0 and x R n , T ( k x ) = T ( x + + x ) = T ( x ) + + T ( x ) = kT ( x ) and T ( x ) = T ((1/ l ) x + + (1/ l ) x ) = T ((1/ l ) x ) + + T ((1/ l ) x ) = lT ((1/ l ) x ), or (1/ l ) T ( x ) = T ((1/ l ) x ) . Also, T ( 0 ) = T ( 0 + 0 ) = T ( 0 ) + T ( 0 ), so T ( 0 ) = 0 . And T ( 0 ) = T ( x + ( 1) x ) = T ( x ) + T ( x ), so T ( x ) = T ( x ).
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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207 - w = f (u) w = f (v) Example: with Example: with where...

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