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hw1extra - EE263 Prof S Boyd EE263 homework 1 additional...

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EE263 Prof. S. Boyd EE263 homework 1 additional exercise 1. A ffi ne functions. A function f : R n R m is called a ffi ne if for any x, y R n and any α , β R with α + β = 1, we have f ( α x + β y ) = α f ( x ) + β f ( y ) . (Without the restriction α + β = 1, this would be the definition of linearity.) (a) Suppose that A R m × n and b R m . Show that the function f ( x ) = Ax + b is a ffi ne. (b) Now the converse: Show that any a ffi ne function f can be represented as f ( x ) = Ax + b , for some A R m × n and b R m . (This representation is unique: for a given a ffi ne function f there is only one
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