05-geometry

05-geometry - 10/6/2003, LINEAR TRAFOS IN GEOMETRY Math...

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Unformatted text preview: 10/6/2003, LINEAR TRAFOS IN GEOMETRY Math 21b, O. Knill Homework to Friday: section 2.2: 4,8,10,30,34,47*,50* LINEAR TRANSFORMATIONS DE- FORMING A BODY A CHARACTERIZATION OF LINEAR TRANSFORMATIONS: a transformation T from R n to R m which satisfies T ( ~ 0) = ~ 0, T ( ~x + ~ y ) = T ( ~x ) + T ( ~ y ) and T ( ~x ) = T ( ~x ) is a linear transformation. Proof . Call ~v i = T ( ~ e i ) and define S ( ~x ) = A~x . Then S ( ~ e i ) = T ( ~ e i ). With ~x = x 1 ~ e 1 + ... + x n ~ e n , we have T ( ~x ) = T ( x 1 ~ e 1 + ... + x n ~ e n ) = x 1 ~v 1 + ... + x n ~v n as well as S ( ~x ) = A ( x 1 ~ e 1 + ... + x n ~ e n ) = x 1 ~v 1 + ... + x n ~v n proving T ( ~x ) = S ( ~x ) = A~x . SHEAR: A = 1 1 1 A = 1 1 1 In general, shears are transformation in the plane with the property that there is a vector ~w such that T ( ~w ) = ~w and T ( ~x )- ~x is a multiple of ~w for all ~x . If ~u is orthogonal to ~w , then T ( ~x ) = ~x + ( ~u ~x ) ~w ....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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