Engineering Calculus Notes 431

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online