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Unformatted text preview: 419 4.1. LINEAR MAPPINGS →
L( − ) =
ı cos α
sin α →
L( − ) = − sin α
cos α so
[ L] = cos α − sin α
sin α cos α . Composition of Linear Mappings
Recall that the composition of two realvalued functions, say f and g, is
the function obtained by applying one of the functions to the output of the
other: (f ◦ g)(x) = f (g(x)) and (g ◦ f )(x) = g(f (x)). For the ﬁrst of these
to make sense, of course, x must belong to the domain of g, but also its
image g(x) must belong to the domain of f (in the other composition, the
two switch roles). The same deﬁnition can be applied to mappings: in
′
′
particular, suppose L: Rn → Rm and L′: Rn → Rm are linear maps (so the
′
natural domain of L (resp. L′ ) is all of Rn (resp. Rn )); then the
composition L ◦ L′ is deﬁned precisely if n = m′ . It is easy to see that this
composition is linear as well:
→
→
→
→
(L ◦ L′ ) α− + β − ′ = L L′ α− + β − ′
x
x
x
x
→
→) + βL′ − ′
′−
x
= L αL ( x
→) + βL L′ − ′
→
′−
= αL L ( x
x
→) + β (L ◦ L′ ) − ′ .
→
′−
= α(L ◦ L )( x
x
It is equally easy to see that the matrix representative [L ◦ L′ ] of a
composition is the matrix product of the matrix representatives of the two
maps:
→
→
(L ◦ L′ )(− ) = L L′ (− )
x
x
′−
= [L] · L (→)
x →
L′ · [− ]
x
→
− ].
′
= [L] · L [ x
= [ L] · We formalize this in ...
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 Spring '08
 ALL
 Calculus

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