Class Notes (836,858)
Canada (509,932)
Mathematics (1,919)
MATH 136 (168)
Lecture 14

# Lecture 14.pdf

4 Pages
59 Views

School
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
More Less

Related notes for MATH 136
Me

Log In

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.

Request Course
Submit