Ka Wai--Chan>>hi TA Bong_Lee>>hi TA Bong_Lee>>do u have enough time to go through the material? Ka Wai--Chan>>no Ka Wai--Chan>>why is Derek so late Siu Him--Lau>>He must be working for prefect society. And I guess Fun is also busy for Green House now. Ka Wai--Chan>>there are only three people TA Bong_Lee>>most holapians are busy? Siu Him--Lau>>Yes. Ka Wai--Chan>>because tomorrow is our swimming gala TA Bong_Lee>>ic Siu Him--Lau>>I am sick today. I can't think of maths now. I am very tired. Ka Wai--Chan>>I think you have a cold Siu Him--Lau>>Yes, for a few days already. Hiu Nga--Cheung>>where is the professor Siu Him--Lau>>He is not always here. TA Bong_Lee>>yes TA Bong_Lee>>otherwise y employ me here? TA Bong_Lee>>any one have question? Ka Wai--Chan>>Eric , you read the book or the notes on the Net Siu Him--Lau>>Both of them. Hiu Nga--Cheung>>Example Fixing an n by n matrix A, matrices X commuting with A are the solutions of the equation AX = XA. The equation is linear because it can be written as AX - XA = O. The left side AX - XA is a linear transformation of X because it is a linear combination of X Õ AX and X Õ XA, which are known to be linear. Hiu Nga--Cheung>>sorry, miss typing Hiu Nga--Cheung>>i don't understand 'Linear Transformation' Siu Him--Lau>>I am sorry that i am going now. I must sleep earlier. Good bye everyone. TA Bong_Lee>>bye TA Bong_Lee>>so can u specific your question? Hiu Nga--Cheung>>why it is a linear combination of X Õ AX and X Hiu Nga--Cheung>>i mean the example above TA Bong_Lee>>your asking y AX-XA is linear transformation of X? Hiu Nga--Cheung>>no Hiu Nga--Cheung>>y X--->AX Hiu Nga--Cheung>>and y X-->XA TA Bong_Lee>>that example is to explain y T(X)=AX-XA is a linear transformation TA Bong_Lee>>since TA Bong_Lee>>F(X)=AX is linear and TA Bong_Lee>>G(X)=XA is also linear TA Bong_Lee>>their sum T(X) =F(X)+G(X) is also linear TA Bong_Lee>>u don't know y F(X) is linear? Hiu Nga--Cheung>>what does it mean by X-->AX TA Bong_Lee>>it just mean if u input X u get back AX TA Bong_Lee>>just a mapping! Hiu Nga--Cheung>>input to where Hiu Nga--Cheung>>T(x)? TA Bong_Lee>>if u input X to F than u get AX Hiu Nga--Cheung>>case V = Rn, W = Rm, Hom(Rn, Rm) can be naturally identified with the collection M(m, n) of m by n matrixes Hiu Nga--Cheung>>why isn't M(n,m) Hiu Nga--Cheung>>The transformation Hiu Nga--Cheung>>D(f) = f': C1(R) Õ C(R) TA Bong_Lee>>Hom(Rn,Rm) is collection of linear transformation from Rn to Rm TA Bong_Lee>>so it map a vector of n-tuple to m tuple TA Bong_Lee>>so we need a matrix with n columns and m rows TA Bong_Lee>>so M should be m by n TA Bong_Lee>>u can try it on paper Hiu Nga--Cheung>>why The transformationD(f) = f': C1(R) Õ C(R) of taking derivatives is linear TA Bong_Lee>>bcos TA Bong_Lee>>d(f+g)/dx=df/dx + df/dx TA Bong_Lee>>and TA Bong_Lee>>d(cf)/dx =c df/dx TA Bong_Lee>>where c is scalar TA Bong_Lee>>satisfy requirement of linear transformation Hiu Nga--Cheung>>df/dx is linear>? TA Bong_Lee>>yes TA Bong_Lee>>where f,g are C1 function Hiu Nga--Cheung>>C1(R) Õ C(R) Hiu Nga--Cheung>>what do they mean Hiu Nga--Cheung>>C(R) is derivative of C1(R)? TA Bong_Lee>>ok TA Bong_Lee>>D:C1(R)->C(R) define D(f)=df/dx TA Bong_Lee>>this is the way we describe a mappping or function TA Bong_Lee>>it tell us that the function map real first differentiable function to continuous function TA Bong_Lee>>and D(f)=df/dx tell us how to map TA Bong_Lee>>clear? Hiu Nga--Cheung>>then what the difference /C1(R) C(R ) Hiu Nga--Cheung>>C1 is differentiable function Hiu Nga--Cheung>>?? TA Bong_Lee>>C1(R) is set of real function which with continuous first derivative TA Bong_Lee>>C(R) just set of continuous real function TA Bong_Lee>>eg TA Bong_Lee>>f(x)=absolute value of x TA Bong_Lee>>is in C(R) but not in C1(R) Hiu Nga--Cheung>>ok Hiu Nga--Cheung>>E(f) = (f(0), f(1), f(2)): F(R) Õ R3 Hiu Nga--Cheung>>R3 means it has 3 element? TA Bong_Lee>>R3 means each of element has three entries or tuples TA Bong_Lee>> Hiu Nga--Cheung>>F(R) is element? TA Bong_Lee>>element of what? Hiu Nga--Cheung>>E(f) TA Bong_Lee>>oh u r confuse TA Bong_Lee>>E(f) is a mapping which map a element in F(R) to R3 TA Bong_Lee>>when we talk about element TA Bong_Lee>>it make sense the element belong to a set not a mapping Hiu Nga--Cheung>> Hiu Nga--Cheung>>E(f) = (f(0), f(1), f(2)): F(R) Õ R3 Hiu Nga--Cheung>>what does it actualy mean TA Bong_Lee>>the statement means TA Bong_Lee>>the function E map element in set of real function to element in R3 by following method TA Bong_Lee>>E(f) = (f(0), f(1), f(2)) where f in F(R) TA Bong_Lee>>clear? Ka Wai--Chan>>sorry I quit now bye Hiu Nga--Cheung>>ok Hiu Nga--Cheung>>are we going to have test soon TA Bong_Lee>>quiz or test? Hiu Nga--Cheung>>Quiz TA Bong_Lee>>on coming sat Hiu Nga--Cheung>>only on chp 2 TA Bong_Lee>>i think so TA Bong_Lee>>but u better send email to Pro Hiu Nga--Cheung>>ok i have to quit now