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log9.pdf

Consider \(T:V\rightarrow V\) and \(S:V\rightarrow V\) are linear transformations on a vector space V. \(S\circ T:V\to V\) is the composition, defined as \(S\circ T(v)=s(T(v))\) , and is also a linear transformation (verify you understand why). \(\mathrm{Ker}(T)\) is the kernel or the null space, defined as

\[ \operatorname{Ker}(T)=\{v\in V\,:\,T(v)=0\}. \]

\(\mathrm{Im}(T)\) is the image, defined as

\[ \operatorname{Im}(T)=\{T(v)\in V\ :\ v\in V\}. \]

Prove or disprove the following claims:

1. If \(\mathrm{Ker}(S\circ T)=\{0\}\) and \(T\) is surjective, then \(\mathrm{Ker}(S)=\{0\}\) and \(\mathrm{Ker}(T)=\{0\}\) .

   Since \(\mathrm{Ker}(S\circ T)=\{0\}\), then \(\operatorname{Ker}(S\circ T)=\{v\in V\,:S\circ\,T\left(v)=0\}=\{v\in V\,:S\left(T(v\right)\right)=0\}=\left\lbrace0\right\rbrace\)

   Since \(T\) is surjective, then \(T(v)=v^{\prime},\forall v^{\prime}\in V\), then \(\operatorname{Ker}(S\circ T)\left.=\{v^{\prime}\in V\,:S\left(v^{\prime}\right.\right)=0\}=\left\lbrace0\right\rbrace\) which means \(v'=0\)​

   Thus \(\text{Ker}(S)=\{0\}\), and \(v'=T(v)=0\Rightarrow v=0\)​

   Thus \(\mathrm{Ker}(T)=\{0\}\)

2. If \(\operatorname{Im}(S\circ T)=\{0\}\) , then \(\operatorname{Ker}(S)=V\) or \(\operatorname{Ker}(T)=V\) .

   Since \(\operatorname{Im}(S\circ T)=\{0\}\), then \(\operatorname{Im}(S\circ T)=\{S\left(T(v\right))\in V:v\in V\}=\left\lbrace0\right\rbrace\)​

   Then \(S(T(v))=0\Rightarrow T\left(v\right)=0\text{ or }S(v^{\prime})=0\), thus \(\operatorname{Ker}(S)=V\) or \(\operatorname{Ker}(T)=V\)​

3. If \(\operatorname{Im}(S\circ T)=\{0\}\) , then \(\operatorname{Im}(T)\subseteq\operatorname{Ker}(S)\) .

   Since \(\operatorname{Im}(S\circ T)=\{0\}\), then \(\operatorname{Im}(S\circ T)=\{S\left(T(v\right))\in V:v\in V\}=\left\lbrace0\right\rbrace\)

   Then \(S(T(v))=0\Rightarrow T\left(v\right)=0\text{ or }S(v^{\prime})=0\), thus \(\operatorname{Ker}(S)=V\) or \(\operatorname{Ker}(T)=V\)

   If \(\operatorname{Ker}(S)=V\), then we know \(\text{Im}(T)\subseteq V=\operatorname{Ker}(S)\)​

   If \(\operatorname{Ker}(T)=V\), then \(\operatorname{Im}(T)=0\subseteq\) any set, thus \(\operatorname{Im}(T)\subseteq\operatorname{Ker}(S)\) in particular

4. Rate the dififculty of the given problem. (1 - very easy, 9 - very hard)

   1 2 3 4 5 6 7 8 9

   What helped you solve the problem?

   Just the definition and some properties taught on lecture.

5. Suppose the problem is given in the exam, and is worth 20 points. How many points do you think you gained for your solution? Please explain.

   18

   Overall, my solution is correctly use definitions, but I think there are some explanation needed to be written to give a more clear solution

6. Do you agree with the following statements? (1 - don’t agree at all, 9 - strongly agree)

(a) The purpose of the problem is to evaluate algorithmic skills, such as arithmetic calculations, technical skills, solving methods, etc.

1 2 3 4 5 6 7 8 9

(b) The purpose of the problem is to evaluate cognitive skills, such as intuitive understanding, train of thought, strategy, etc.

1 2 3 4 5 6 7 8 9

(c) The purpose of the problem is to evaluate formal skills, such as mathematical logic, proof structure, proper writing, etc.

1 2 3 4 5 6 7 8 9

1. (a) Does your experience in GTIIT so far meet you expectations of how you thought it would be? Can you give an example of something that surprised you, and wasn’t as you previously expected?

   No, The public office hour by lecturer is very short, and it is hard to appoint the office hour to lecture. One reason is my schedule is full and hard to have times to go to the office hour. Another reason is the professor Paulo seem unwilling to give more office hour because he usually leave the office before the end of public office hour. I don't know why...

(b) Supposed you were asked to give an advise to a new student starting their studies in the MCS program in GTIIT, what advise would you give them?

Think twice, third, fourth,... think many times. Be sure you have the confident and ambition to conquer any difficulties.

(c) Do you enjoy your time in GTIIT so far? (1 - I don’t enjoy at all, 9 - I enjoy a lot)

1 2 3 4 5 6 7 8 9

Please elaborate: I think my time is full of study, however i enjoy this status since i don't like having too much social communications and i don't like to go out to play or eating delicious food. I think it is boring, what i enjoyed is the process of progress although it is hard and do some funny staffs about CS.