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