## 3B1B: Essence of Linear Algebra

### 1: Vectors

3 main ideas surrounding vectors.

- vectors - arrows pointing in space - has length and directon position does not matter can be 2D and 3D.
- vectors - ordered list of numbers - i.e vector fancy word for list

last one i dont resonate with but here it is regardless. i dont really *get* it ig.

- vectors - can be anything as long as there is an addition and a multiplication operation possible.(Multiplication by a number). ok this is abstracting away from the representation. that is its purpose.

**think of vectors as the 1st idea in a coordinate system with its tail(the part without the arrow) rooted at the origin**

the goal is to understand concepts using this coordinate plane idea and then transfer that to the list of numbers idea.

know this idea already. how to represent vectors in a cordinate plane.

[coordinateAxe1 coordinateAxe2 coordinateAxe3 coordinateAxe4 …] -> Number of coordinate Axes -> Dimensions

[x y] -> 2D

[x y z] -> 3D

this addition of vectors visualization is cool.

### COOL IDEA ALERT!!

think of vectors as movement in space with a direction and a magnitude.

same idea can be applied to the number line obv

[x_{1} y_{1}] + [x_{2} y_{2}] = [x_{1}+x_{2} y_{1}+y_{2}]

multiplication of a vector by a number is streching the vector if the number is >1, shrinking the vector if the number is <1 and the same for negative numbers but the vector is flipped first.

this operation can be called scaling (for obvious reasons) and the number which scales the vector is called a ** scaler**!

throughout linear algebra one of the most common things numbers do is scale vectors. so scaler <-> number

2[x y] -> [2x 2y]