Linear Algebra Lecture: Vector Addition and Linear Combinations

Jul 26, 2024

Linear Algebra Lecture Notes

Introduction

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

Key Basic 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).

Linear Combination

  • 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 ]

Changing Basis Vectors

  • 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 of Vectors

  • 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.

Visualizing Vectors

  • 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.

Three-Dimensional Vectors

  • 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.

Linear Dependence and Independence

  • 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.

Conclusion

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

Next Steps

  • Upcoming video: introduction to matrices and transforming space.