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.

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