Jul 26, 2024

**Vector Coordinates**: Relationship between pairs of numbers and two-dimensional vectors.- Think of coordinates as scalars that stretch or compress vectors.

**Unit Vectors**:**i-hat**(x-direction, right, length 1)**j-hat**(y-direction, up, length 1)

- Each coordinate in a vector scales these unit vectors:
- x coordinate scales i-hat (e.g., stretching by a factor of 3).
- y coordinate scales j-hat (e.g., flipping and stretching by a factor of 2).

- Combining scaled vectors:
- The vector represented by coordinates (3, -2) = 3 * i-hat + (-2) * j-hat.

**Basis Vectors**: Basis of a coordinate system consists of these unit vectors (i-hat and j-hat).**Linear Combination**: Sum of scaled vectors, described mathematically by: [ a \cdot , \textbf{v}_1 + b \cdot , \textbf{v}_2 ]

- Possible to choose different basis vectors that still create a valid coordinate system.
- Altering scalar choices alters the vectors:
- All combinations of two scalars can represent every possible two-dimensional vector, unless the vectors are collinear.

**Span**: Set of all possible vectors reachable by linear combinations of a pair of vectors.- For most pairs of 2D vectors, the span is the entire 2D space.
- If vectors are collinear, the span is limited to a line.

**Points vs Vectors**:- Single vectors are represented as arrows.
- Collections of vectors can be represented as points in space.

- Thinking about spans visually:
- Span of two vectors in 3D will create a flat sheet.

- Adding a third vector changes the span:
- If not collinear with the first two, it provides access to all 3D vectors.
- If it is collinear, the span remains unchanged.

**Linearly Dependent**: At least one vector can be expressed as a linear combination of others, forming overlapping spans.**Linearly Independent**: All vectors add new dimensions to the span.

**Basis Definition**: A set of linearly independent vectors that spans a space.- Importance of understanding span and linear independence in linear algebra.

- Upcoming video: introduction to matrices and transforming space.