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log3.pdf

Prove or disprove the following claim: Let \(V\) be a vector space over a field \(\mathbb{F}\), and let \(U, W\) be vector subspaces of \(V\). If \(U \cup W\) is a vector subspace of \(V\) then \(U \subseteq W\) or \(W \subseteq U\).

Proof

\(\Rightarrow\)) Take \(u\in U\) and \(w\in W\), since \(U\subseteq U\cup W,W\subseteq U\cup W\), then \(u\in U\cup W,w\in U\cup W\)

Thus \(u+w\in U\cup W\), then we have two cases

If \(u+w\in U\), since \(u\in U\), then \(-u\in U\), then \(u+w+\left(-u\right)\in U\Rightarrow\) \(w\in U\)

Thus we have \(W\subseteq U\)

If \(u+w\in W\), since \(w\in W\), then \(-w\in W\), then \(u+w+\left(-w\right)\in W\Rightarrow\) \(u\in W\)

Thus we have \(U\subseteq W\)

\(\Leftarrow\)) If \(U_{}\subseteq W\Rightarrow U\cup W=W\) and since \(W\) is a subspace, so is \(U\cup W\).

If \(W\subseteq U\Rightarrow U\cup W=U\), which is a subspace.

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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. How confident are you that you answered correctly? (1 - not confident at all, 9 - very confident)

   1 2 3 4 5 6 7 8 9

   • If you think you were right, what helped you answer correctly?

   Actually, i see this problem in the tutorial. Then i review this question after class, then i get a deeper understanding of this problem. So i think i definitely is right * If you think you were wrong, what could have helped you answer correctly?

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

   (a) I understand the problem’s instructions and what it means to write a formal solution. 1 2 3 4 5 6 7 8 9

   (b) I intuitively understand what is the solution to the problem. 1 2 3 4 5 6 7 8 9

   (c) Intuition is important in order to be good in mathematics. 1 2 3 4 5 6 7 8 9

4. What are the differences between learning mathematics in GTIIT and learning mathematics in high school?

   The difficulty of learning and a diverse array of courses. Sometimes i may find that i have a lot of notes and exercises to do and many books needed to read. Also, our courses are hard, we usually study deeper and deeper. Not stay on the surface

   What do you do in order to overcome these differences?

   Review my notes and repeat it to myself to examine whether i have a good command of it. I selects some important part to read instead of reading whole books. And I don't do many exercises but try to cultivate my generalization ability.