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Unformatted text preview: EE263 Prof. S. Boyd EE263 homework 1 additional exercise
1. Aﬃne functions. A function f : Rn → Rm is called aﬃne if for any x, y ∈ Rn and
any α, β ∈ R with α + β = 1, we have
f (αx + β y ) = αf (x) + β f (y ).
(Without the restriction α + β = 1, this would be the deﬁnition of linearity.)
(a) Suppose that A ∈ Rm×n and b ∈ Rm . Show that the function f (x) = Ax + b is
aﬃne.
(b) Now the converse: Show that any aﬃne function f can be represented as f (x) =
Ax + b, for some A ∈ Rm×n and b ∈ Rm . (This representation is unique: for a
given aﬃne function f there is only one A and one b for which f (x) = Ax + b for
all x.)
Hint. Show that the function g (x) = f (x) − f (0) is linear.
You can think of an aﬃne function as a linear function, plus an oﬀset. In some
contexts, aﬃne functions are (mistakenly, or informally) called linear, even though in
general they are not. (Example: y = mx + b is described as ‘linear’ in US high schools.) 1 ...
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This note was uploaded on 01/25/2012 for the course EE 263 taught by Professor Boyd,s during the Spring '08 term at Stanford.
 Spring '08
 BOYD,S

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