Homework 1
For all the following exercises, if you feel it is too easy, skip it; if you find it interesting and relevant, do it; if you find it too hard, ask about it on discord, let's tackle it together.
1. If you are not familiar with set theory notation and terminology, watch these two short videos: 1. set notations , 2. how to describe a set (5 min each). If you need to know what is injective / surjective of map of sets, try wikipedia.
- Explain in your own word, what is a set, what is a map. Give examples.
- Can you put 'unrelated things' in a set? Like, a set that consist of three elements: $\{\text{your instructor Peng}, \text{a Gorilla}, \text{the Sun}\}$?
- When you define a map $f: A \to B$, where $A,B$ are two sets, can one element in $A$ be sent to two or more elements in $B$?
2. About linear map. The following functions $f: \R \to \R$, which is a linear map, and which is not?
- $f(x) = |x|$
- $f(x) = (x)^2$
- $f(x) = 3x$
- $f(x) = x+1$.
3. About linear subspace. Let $V = \R^2$, is the following subset $V' \In V$ a subspace? Explain why. You need to check if elements in $V'$ are closed under the vector addition and scalar multiplication. (if you are unfamiliar with the set notation, watch the two videos above.)
- $V' = \{ (x,y) \in \R^2 \mid x+y=0 \}$
- $V' = \{ (x,y) \in \R^2 \mid x+y=1 \}$
- $V' = \{ (x,y) \in \R^2 \mid x>0, y>0 \}$.
4. Matrix manipulations. Write out a $3 \times 2$ matrix $A$, and a $2 \times 2$ matrix $B$, and multiply them together $AB$. Does $BA$ make sense? What do you get?
5. Equivalence relation and equivalence classes. read about it on wiki. Explain it in your own words and give examples. https://en.wikipedia.org/wiki/Equivalence_class
6. Ask someone in your class a question. (the more question the better) Write down the question that you asked and who you asked (could be as simple as “what is $\forall$?” I hope to use this as an excuse to open up talking). You can DM people on discord.
Homework is due on Monday in class. Please write or print out your answer. The homework is graded by completion. It is good opportunity to make mistakes. If you don't do it (not because of it's easy but because you are lazy), then you don't get points.
