Engineering Calculus Notes 431

Engineering Calculus Notes 431 - 419 4.1 LINEAR MAPPINGS...

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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 real-valued 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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