Lecture Notes on Vectors and Linear Algebra

Introduction

Vector coordinates can be thought of as scalars that stretch or squish vectors.
In the xy coordinate system:
- i-hat: Unit vector in the x-direction (points right, length 1).
- j-hat: Unit vector in the y-direction (points up, length 1).
Example: For a vector (3, -2):
- x-coordinate (3) scales i-hat, stretching it by a factor of 3.
- y-coordinate (-2) scales j-hat, flipping and stretching it by a factor of 2.
The resultant vector is a sum of these scaled vectors.

Basis Vectors: i-hat and j-hat are the basis of the coordinate system.
Different basis vectors can create new coordinate systems.
- Choose two vectors: they define a new system.
Linear Combination: Combining two vectors with scalars.
- Fix one scalar, let the other vary will result in a straight line.
- Allow both scalars to vary can cover all 2D vectors unless they are collinear.

Span: Set of all possible vectors reachable through linear combinations of two vectors.
- Most 2D vectors span the entire 2D space.
- Collinear vectors only span a line.
Linear algebra focuses on vector addition and scalar multiplication.

Points vs. Arrows:
- Single vector: conceptualize as an arrow.
- Collection of vectors: represent as points (tip of the vector).
Span of two vectors (non-collinear): flat sheet in 3D space.

Adding a third vector to the span:
- If it lies on the span of the first two, no new vectors are added.
- If it's independent, it allows access to all 3D vectors.
Linear Dependence: A vector is dependent if it can be formed from a combination of others.
- Vectors can be linearly independent if they add new dimensions.