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Lecture 14

Lecture 14.pdf

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Department
Mathematics
Course
MATH 136
Professor
Robert Sproule
Semester
Winter

Description
Wednesday, February 5 − Lecture 14 : Linear mappings (transformations) I Concepts: 1. linear mapping (transformation) 2. matrix viewed as a linear mapping 14.1. Definition − A linear mapping (also called “linear transformation”) is a function T : ℝ → ℝ where, for any pair of vectors uand v in ℝ and scalars α and β, T(αu + βv) = αT(u) + βT(v) We sometimes say that T is linear if it “respects linear combinations”. Note that some authors separate the definition of a linear transformation in two parts: 1) T (αx) = αT (x) (We say that T “preserves” scalar multiples.) 2) T(x + y) = T(x) + T(y) (We say that T “preserves” addition.) These definitions are equivalent. 14.1.1 Example – We define the function T: ℝ → ℝ , as 2 T [ (x , x , x )] = (x , x ) 1 2 3 1 2 For example, T [ (1, 5, 2) ]= (1, 5). This is a linear mapping since T [α(x1, x2, x3) + β(y 1 y2, y3)] = T [(αx 1 αx 2 αx 3 + (βy ,1βy ,2βy )3 ] = T [(αx 1 βy ,1αx + 2y , αx2+ βy 3 3 ] = (αx +1βy , 1x + 2y ) 2 = (αx 1, αx2) + (βy , 1y ) 2 = α(x ,1x )2+ β(y , y 1 2 = αT [(x1, x2, 3 ) ] + βT [(y 1 y2, y3)] 14.2 The matrix A m × niewed as a linear mappingfrom ℝ into ℝ . m Given a matrix A m × n= [aij m × n with rows r,iwe define a function T : ℝ → ℝ as m T(x) = Ax. In fact T : ℝ → ℝ is a well-defined function. We claim it is a linear mapping. n For two vectors x and y in ℝ and any scalars α and β we have: - Hence the matrix A m × ncan be viewed as a functionmapping a vector x in ℝ to a n vector Ax in ℝ .m - We have also shown that matrices distributes over a linear combination. That is
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