Hiu Nga--Cheung>>i have post question on discussion board
TA Bong_Lee>>how about your quiz2?
Hiu Nga--Cheung>>terrible
Hiu Nga--Cheung>>The subset (see this exercise)
Hiu Nga--Cheung>>{f F(R): f(1) = f(2) = 0} â F(R)
Hiu Nga--Cheung>>Kernel and Null Space
Hiu Nga--Cheung>>part 1
TA Bong_Lee>>what's the Q?
Hiu Nga--Cheung>>example1
Hiu Nga--Cheung>>{f F(R): f(1) = f(2) = 0} â F(R)
Hiu Nga--Cheung>>is a subspace because it is the kernel of the linear transformation
Hiu Nga--Cheung>>E(f) = (f(1), f(2)) F(R) Õ R2.
Hiu Nga--Cheung>>why it is F(R)-->R2
TA Bong_Lee>>bcos
TA Bong_Lee>>bcos
TA Bong_Lee>>if u input a fuction in real number than you grt back (f(1),f(2)) which is element in R2
TA Bong_Lee>>for example
TA Bong_Lee>>u input f(x)=1/x
TA Bong_Lee>>u get back(1,1/2) which in R2
Hiu Nga--Cheung>>why you get back (1,1/2)?
TA Bong_Lee>>bcos
TA Bong_Lee>>f(x)=1/x and (f(1),f(2))=(1,1/2)
TA Bong_Lee>>by the way i want to ask are holapian busy?
Hiu Nga--Cheung>>i don't understand
Hiu Nga--Cheung>>holapian
Hiu Nga--Cheung>>?//
TA Bong_Lee>>u r not come from HOLAP school?
Hiu Nga--Cheung>>no
Siu Him--Lau>>i am
TA Bong_Lee>>quiz ok?
Siu Him--Lau>>i hope i can get a pass
Hiu Nga--Cheung>>isn't f(1) = f(2) = 0
Hiu Nga--Cheung>>how can you get (f(1),f(2))=(1,1/2)
TA Bong_Lee>>I think u r not clearly understand the description of a mapping
TA Bong_Lee>>let me explain
TA Bong_Lee>>let M be mapping map function in real number to element in R2
TA Bong_Lee>>so we write M: f(R)->R2
TA Bong_Lee>>we should tell people how this can be done
TA Bong_Lee>>so we define M by M(f(R))=(f(1),f(2))
TA Bong_Lee>>so u input a function in real number,you take the value of the function at 1 and 2
TA Bong_Lee>>now the kernel of the mapping M is the set of function in real number with the property
TA Bong_Lee>>f(1)=0=f(2)
TA Bong_Lee>>so
TA Bong_Lee>>{f(x) in F(R) :f(1)=0=f(2)} is kernel of M
TA Bong_Lee>> clear?
TA Bong_Lee>>I think all of u should have familarized the description statement of a mapping
Hiu Nga--Cheung>>that means there is a function satisfy f(1)=0=f(2)
TA Bong_Lee>>otherwise u will have problem in coming midterm
TA Bong_Lee>>of course
Hiu Nga--Cheung>>the function is called kernel
TA Bong_Lee>>no
TA Bong_Lee>>Kernel is a set
TA Bong_Lee>>set of function
Hiu Nga--Cheung>>then there is some function doesn't meet f(1)=0=f(2
TA Bong_Lee>>yes
Hiu Nga--Cheung>>Kernel is a set of function which all suit f(1)=0=f(2
TA Bong_Lee>>right!
TA Bong_Lee>>but Kernel for the mapping M only
TA Bong_Lee>>not all mapping!
Hiu Nga--Cheung>>yes
Hiu Nga--Cheung>>M(f) = (t2+1)f: F(R) Õ F(R)
Hiu Nga--Cheung>>Therefore kernelM = {0} and M is one-to-one.
TA Bong_Lee>>what is M(f)=(t2+1)?
Hiu Nga--Cheung>>the kernel is set of f that (t2+1)f=0
TA Bong_Lee>>ok ic
Hiu Nga--Cheung>>the last example in part 1
TA Bong_Lee>>yes
Prof Yan--Min-->>hello everybody
TA Bong_Lee>>the kernel={f(x)=0}
Hiu Nga--Cheung>>f=0 why means one to one
TA Bong_Lee>>one element only
Hiu Nga--Cheung>>how can i know it is one element only
TA Bong_Lee>>let me answer your 1st q first
TA Bong_Lee>>for linear transformatiuon
TA Bong_Lee>>T(a+b)=T(a)+T(b)
TA Bong_Lee>>if T(a)=T(b)
TA Bong_Lee>>then T(a)-T(b)=0
TA Bong_Lee>>T(a)+T(-b)=0
TA Bong_Lee>>T(a-b)=0
TA Bong_Lee>>and if kernel only contain 0
TA Bong_Lee>>than a-b must =0
TA Bong_Lee>>imply a=b
TA Bong_Lee>>so T is one to one
TA Bong_Lee>>and remember 0 must in kernel of liner transformation
TA Bong_Lee>>so kernel contain only one element is equivalent to say that ker={0}
TA Bong_Lee>>anything miss Pro Yan?
Prof Yan--Min-->>your answer is very good. more Q please
Hiu Nga--Cheung>>If we change t2 + 1 to t2 - 1,
Hiu Nga--Cheung>>M is no longer onto why
Prof Yan--Min-->>Because M(f) = (t^2-1)f(t) must be zero at t =1. Thus g(t) = t is not of the form M(f).
Prof Yan--Min-->>In other words, g(t) = t is not in the range of M(f)
Prof Yan--Min-->>Is my answer OK?
Hiu Nga--Cheung>>what is g(t)
Prof Yan--Min-->>g(t) is the function t
Prof Yan--Min-->>you may also take g(t) = t + 1 or g(t) = 2t and still get the same conclusion
Hiu Nga--Cheung>>what is the form M(f)
Prof Yan--Min-->>M(f) = (t^2 -1)f(t). This is the definition.
Prof Yan--Min-->>For example, M(1) = t^2-1, M(t) = t^3 - t, M(sint) = t^2sint - sint, etc.
Prof Yan--Min-->>more Q?
Hiu Nga--Cheung>>if we restrict the source and target by considering M(f) = (t2-1)f: C(R) Õ C(R),
Hiu Nga--Cheung>>what is C(R)
Prof Yan--Min-->>the vector space of all continuous functions (on the domain R)
Hiu Nga--Cheung>>why the linear transformation is again one-to-one
Prof Yan--Min-->>For general function f, (t^2 -1)f(t) = 0 does not imply f(t) = 0
Prof Yan--Min-->>For continuous function f, (t^2 - 1)f(t) = 0 implies f(t) = 0
Prof Yan--Min-->>Why is only Hiu Nga asking Qs?
Prof Yan--Min-->>This does not mean Hiu Nga cannot ask more Qs
Hiu Nga--Cheung>>could you see the discusion board
Prof Yan--Min-->>OK I am opening it now.
Prof Yan--Min-->>OK. I am reading your first Q on M(f) = (t^2 + 1) f(t).
Prof Yan--Min-->>You repeated my example but did not ask Qs
Prof Yan--Min-->>You second Q: why is T(f) = f(1) + f(2): F(R) -> R onto?
Prof Yan--Min-->>The keyb is to understand the meaning.
Prof Yan--Min-->>T is onto means that for any number a, we can find a funciton f, such that f(1) + f(2) = a.
Prof Yan--Min-->>Understand?
Prof Yan--Min-->>Repeat: a number a is in the range <=> we can find a funciton f, such that f(1) + f(2) = a
Hiu Nga--Cheung>>rangeT=R what is R
Hiu Nga--Cheung>>real no
Hiu Nga--Cheung>>??
Prof Yan--Min-->>Yes
Prof Yan--Min-->>More Q on your post on the discussion board?
Hiu Nga--Cheung>>F(R)-->R
Prof Yan--Min-->>V -> W means a transformation with domain/source V and target W
Prof Yan--Min-->>T(f) = f(1) + f(2) takes a function f in F(R) to a number f(1) + f(2) in R. Thus T is F(R) -> R
Prof Yan--Min-->>Did I answer your Q?
Hiu Nga--Cheung>>example
Prof Yan--Min-->>T(f) = f(1) + f(2) takes f(t) = t + 1 in F(R) to f(1) + f(2) = 2 + 3 = 5 in R
Prof Yan--Min-->>It seems you are confused about notations.
Hiu Nga--Cheung>>yes
Prof Yan--Min-->>Perhaps you should read the appendix (last section of the chapter) on transformations
Hiu Nga--Cheung>>i don't understand the difference /T and T(f)
Prof Yan--Min-->>T is the notation for the transformation
Prof Yan--Min-->>T(f) is what you get by applying the transformation to a vector f
Hiu Nga--Cheung>>F(R)?
Prof Yan--Min-->>F(R) is different from T(f)
Prof Yan--Min-->>Unfortunately!
Prof Yan--Min-->>F(R) is ONE notation, meaning Functions defined on R
Prof Yan--Min-->>F stands for function, R stands for the real numbers (the domain of the functions)
TA Bong_Lee>>i have to go bye
Hiu Nga--Cheung>>like f(x)
Prof Yan--Min-->>T(f) is a combination notation
Prof Yan--Min-->>T(f) is like f(x)
Prof Yan--Min-->>Notation-wise
Prof Yan--Min-->>f is the notation for the funciton, and x is the notaiton for the variable
Prof Yan--Min-->>F(R) is not f(x)
Prof Yan--Min-->>F(R) is the collection/set of all functions
Prof Yan--Min-->>F(R) = {f(x): f is a real-valued function defined for x in R}
Prof Yan--Min-->>F(R) is made up of (many many different) f(x). It is not equal to f(x)
Prof Yan--Min-->>R^3 is made up of (1, 2, 2), (0, 1, -2), (2, 0, 0), etc.
Prof Yan--Min-->>But R^3 is not equal to (1, 2, 2), not equal to (0, 1, -2)
Hiu Nga--Cheung>>any example of F(R)
Prof Yan--Min-->>Can you tell me any example of R^3?
Hiu Nga--Cheung>>(5,8,9)
Prof Yan--Min-->>F(R) is the set of all functions, examples of ELEMENTS of F(R) are t, t^2, sint, e^t, etc.
Prof Yan--Min-->>Strictly speaking, (5, 8, 9) is an example of VECTORS of R^3.
Prof Yan--Min-->>(5, 8, 9) is not an example of R^3 itself
Hiu Nga--Cheung>>how about (x,y,z)
Prof Yan--Min-->>Now this is controversal
Prof Yan--Min-->>What do you mean (x, y, z)?
Hiu Nga--Cheung>>x can be no/function
Prof Yan--Min-->>For a specific choice of numbers x, y, and, z, (x, y, z) is an example of vector of R^3
Prof Yan--Min-->>If you allow all the arbitrary choices of x, y, z, then it IS R^3 it self, not just AN EXAMPLE
Hiu Nga--Cheung>>i think i get it now
Prof Yan--Min-->>By the way, only for numbers (x, y, z) is a vector in R^3.
Prof Yan--Min-->>(t, t^2, t^3) should not be considered as a vector of R^3.
Prof Yan--Min-->>For each specific value of t, (t, t^2, t^3) is a vector in R^3
Hiu Nga--Cheung>>i fall behind the schedule
Prof Yan--Min-->>If you consider t, t^2, t^3 as funcitons (i.e., allow arbitrary t), then (t, t^2, t^3) should be considered as a vector in F(R)^3
Prof Yan--Min-->>Note that R in R^3 stands for real numbers.
Prof Yan--Min-->>3 in R^3 stands for THREE (real numbers)
Prof Yan--Min-->>How far have you been up to?
Hiu Nga--Cheung>>range
Prof Yan--Min-->>let me see the schedule
Prof Yan--Min-->>Are you very busy recently?
Hiu Nga--Cheung>>very busy
Prof Yan--Min-->>on what?
Hiu Nga--Cheung>>tests
Prof Yan--Min-->>midterm?
Hiu Nga--Cheung>>no
Prof Yan--Min-->>do you have midterm in high school?
Prof Yan--Min-->>What kind of tests?
Hiu Nga--Cheung>>yes we have but mark of test is included in report
Hiu Nga--Cheung>>test on physic and chemistry pure maths and chinese
Prof Yan--Min-->>do you want to call me tomorrow? typing conversation is no fun
Hiu Nga--Cheung>>yes
Hiu Nga--Cheung>>i will use phone next time
Prof Yan--Min-->>my office number is 23587442. Please try to call me tomorrow morning.
Prof Yan--Min-->>I will be busy tomorrow afternoon
Hiu Nga--Cheung>>i have school tomorrow
Prof Yan--Min-->>when can you make phone call?
Hiu Nga--Cheung>>night
Hiu Nga--Cheung>>or saturday morning
Prof Yan--Min-->>Please specify a time on saturday morning. I will make sure I am in my office then
Hiu Nga--Cheung>>11:00
Prof Yan--Min-->>I just remember I have to leave at 10:30
Prof Yan--Min-->>Can you call mebefore 10?
Hiu Nga--Cheung>>then9:30
Prof Yan--Min-->>OK. I will be in my office at 9:30 on Saturday. Bye
Hiu Nga--Cheung>>bye