====== Review of Linear Algebra ======

Let's start from scratch again. What is linear algebra? 

This is a [[https://math.berkeley.edu/~giventh/la.pdf | textbook on linear algebra]] by Prof Givental. 


===== What is a vector space =====
==== Answer 1 ====
row vectors, column vectors, matrices. Let's also review the index notation $a_i = \sum_{j} M_{ij} b_j$

Very concrete, very computable. 


==== Answer 2 ====
Geometrical, as we went over in class. 

A 2-dimentional vector is something you can draw. 

A 3-dim vector, hmm, harder. 

how about 4-dim vector? $\infty$-dim one? It doesn't matter the dimension, the rule we obtain from 2 and 3 dimensional one is good enough. 

==== Answer 3 ====
the goofy math prof: a vector space is a set $V$ together with two operations
  * scalar multiplication: given a number $c$ and a vector $v \in V$, we need to specify the output $c v \in V$. 
  * vector addition: given two vectors $v_1, v_2 \in V$, we need to specify the output $v_1 + v_2 \in V$
such that, some obvious conditions should be satisfied. 

Why we care about this? Because it is somehow useful.

For example, 
  * the subspace of $\R^3$ that is perpendicular to $(1,2,3)$ forms a vector space.
  * other examples? non-examples? 


===== Linear Map =====
How does vector space talk to each other? Linear map. 

Do an example of stretching, skewing. 

Do a non-example of bending a line in $\R^2$. 

Kernel, image and cokernel


We didn't quite cover the idea of a quotient space. I will do that next time.