====== 2020-01-22, Wednesday ======

Hi, this is my post-lecture note. A short summary of what we did in class today. (Also a chance to correct my mistakes made in class)

So, what is a vector? There are many possible correct answers (in different sense). You may say one of the following
  * it is an arrow with a starting point and ending point (ha, you are thinking of a //displacement vector//)
  * it is an ordered set (a.k.a tuple) of numbers, like $(2,3,-1)$ a 3-dim vector, or $(2,3)$ a 2-dim vector. You represent it by an arrow from the origin to the point with that coordinate. 
  * (and the most boring yet correct answer) it is an element in the vector space. (well, great, it is like asking, what is 2 + 3? and the answer is 3 + 2.) but this is correct, since, after we learned so many examples of various general vector spaces (the space of degree 2 polynomials, the solution space to differential equations, etc), you cannot simply say that vector is just a tuple of numbers. 
 

The notion of vector space is the following. 

//Let $V$ be a set, we say $V$ is a vector space $V$ over a field $k$ (say, $k=\R$ or $k=\C$) if the following is true
  * we can add two vectors together and get a vector
  * we can multiply a number (in $k$) to a vector and get a vector
  * these two operations needs to satisfy some obvious relations, i.e., if $a, b \in k$ and $v,w \in V$, then $$ (ab)v = a(bv), \quad a(v+w) = av+aw, \quad  a \neq 0, v \neq 0 \Rightarrow av \neq 0 \cdots$$.
//

We then give some examples of vector spaces. 

** Example **: \\
  - Fix a positive number $N$, then the set of real coefficient polynomials $f(x)$, where $deg(f) \leq N$ forms a vector space over $\R$. A basis of this vector space is $\{1, x, \cdots, x^N\}$, hence the dimension is $N+1$. 
  - The space of solution to $$f''(x) + x f(x) = 0$$ forms a 2 dimension vector space. 
  - In general, a degree $k$ homogeneous ordinary differential equation has solution space $k$-dimensional. That is why when we write down the general solution, we write $$ f(x) = c_1 f_1(x) + \cdots + c_k f_k(x)$$ where the set $\{f_i(x)\}$ are linearly independent solutions, and $c_i$ are constants.

OK, now that we see many 'weird' vector spaces, we may think twice when we say, "the length of a vector is just $\sqrt{x^2+y^2+z^2}$", and a vector is just a tuple of numbers. But vector can be identified with a tuple of numbers, once we fix a ...

==== Basis ====

A basis of a vector space $V$ is a maximal collection of linearly independent vectors $V$. The number of vectors in a basis is called the //dimension//. 

Q: is it possible that, for a vector space $V$, I can find a basis with $3$ elements, but someone else can find a basis with $4$ elements? Why not? 


==== Coordinates ====
Given a basis $e_1, \cdots, e_n$ of an n-dimensional vector space $V$, any element $v$ can be identified with a tuple of numbers $(v_1, ..., v_n)$ by the following relation
$$ v = v_1 e_1 + \cdots + v_n e_n $$
These $(v_1, \cdots, v_n)$, or $v_i$s,  are //coordinates// of $v$ with respect of the basis $\{e_i\}$. 

It is no surprise that, if you change the basis, you will also change the coordinates. Just like when you change unit, the number in front of it changes, 2 meter = 6.5 feet (NBA player Michael Jordan's height), even though they both represent the same length. 

That's all. Next time, we will see what is a tensor.