6

log6.pdf

Let \(T:\mathbb{C}^{2\times2}\to\mathbb{C}^{2\times2}\) be defined as \(T\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b - c \\ c - b & a \end{pmatrix}.\)​

1. Prove that \(T\) is linear.

   Consider \(T(v+\lambda u)=T\left(\begin{pmatrix}a & b\\ c & d\end{pmatrix}+\lambda\begin{pmatrix}e & f\\ g & h\end{pmatrix}\right)=T\left(\begin{pmatrix}a+\lambda e & b+\lambda f\\ c+\lambda g & d+\lambda h\end{pmatrix}\right)=\begin{pmatrix}a+\lambda e & b-c+\lambda f-\lambda g\\ c-b+\lambda g-\lambda f & a+\lambda e\end{pmatrix}\)​

   \(T(v)+\lambda T(u)=\begin{pmatrix}a & b-c\\ c-b & a\end{pmatrix}+\lambda\begin{pmatrix}e & f-g\\ g-f & e\end{pmatrix}=\begin{pmatrix}a+\lambda e & b-c+\lambda f-\lambda g\\ c-b+\lambda g-\lambda f & a+\lambda e\end{pmatrix}\)​

   Thus \(T(v+\lambda u)=T(v)+\lambda T(u)\), then this is linear

2. Find a basis and dimension for \(\text{Im}(T)\text{ and }\text{Ker}(T).\)

   \(\text{Im}(T)=\langle\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\rangle\)​

   \(\text{Ker}(T)={}\langle\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}\rangle\)

3. Find a non-zero and non-invertible matrix \(A\) in the image.

   Non-zero: \(\begin{pmatrix}1 & 1\\ -1 & 1\end{pmatrix}\) Non-invertible: \(\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\)​

4. Does the matrix \(\begin{pmatrix} 2 & 3 \\ -3 & 2 \end{pmatrix}\)​belong to the image? If so, find the set of origins, and determine if this set is a vector space. If it is not, explain why.

   Yes, let \(T\begin{pmatrix}a & b\\ c & d\end{pmatrix}=\begin{pmatrix}a & b-c\\ c-b & a\end{pmatrix}=\begin{pmatrix}x & y\\ -y & x\end{pmatrix}\), we have \(a=x,b-c=y\)

   Then the preimage is \(\begin{pmatrix}x & y+c\\ c & d\end{pmatrix}\) where \(c,d\) can be any value

   Let's check \(0\in\) preimage

   \(x=0,y=-c\) but \(c,d\) can be non-zero, thus \(0\notin\) preimage

   Then this is not a vector space

Answer the following questions:

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

   1 2 3 4 5 6 7 8 9

2. Suppose the problem is given an exam, and is worth 20 points.

   How many points do you think you gained for your solution? Please explain.

   10, I didn't understand the meaning of origins, but I guess it is the pre-image, if it is the pre-image, then i have more confident, i think i'm right, because the image of pre-image is exactly same with the image i have solved

3. 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

4. Consider the following quote:The most important resource that you have in the course, by far, are the people around you. It is important that you know each other. Talk to each other, ask each other questions, ask others for feedback."

   Do you agree with the this quote?(1 - don’t agree at all, 9 - strongly agree)

   1 2 3 4 5 6 7 8 9

   Please explain:

   Because talking will disturb my thread and focus to listen the lecture and this action will also disturb others, because this will cause some noise, i don't like do that, it is not polite.

   Usually, i am used to thinking and studying in a silent environment. And also listening to the interpretation is more important than talking with friends.

   Admittedly, Talking to friends after class is a useful way to improve skills.

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