Understanding Basis and Coordinate Systems

Nov 3, 2024

Lecture on Basis and Coordinate Systems in Vector Spaces

Importance of Basis

  • Basis is crucial in the study of vector spaces.
  • For a finite basis, the dimension of a vector space is its size.
  • All bases in a given vector space must have the same size.
    • Example: R3 is three-dimensional with basis E1, E2, E3.

Dimension

  • Dimension provides a crisp and precise definition.
  • Translates abstract ideas into concrete concepts.

Coordinate Systems

  • Describe positions in space using numbers.
  • Example: Cartesian coordinates in the xy-plane.
    • Position described using two numbers (horizontal, vertical).

Cartesian Coordinates Example in R2

  • Standard Cartesian Coordinates:
    • Uses vectors E1 and E2.
    • Example Point V is (4, 3) in standard coordinates.
    • V is 4 copies of E1 + 3 copies of E2.

Non-Standard Cartesian Coordinates

  • Use alternative basis instead of standard basis.
  • Example basis: (1, 2), (3, 1).
    • Forms a grid with parallelograms instead of squares.
    • V can be expressed differently, e.g., as (1, 1) in this basis.

Coordinate System Construction

  • Choose a basis for a vector space V.
  • Element V expressed as a linear combination of basis vectors.
  • Weights (coefficients) are unique due to linear independence.
    • Expressed as a column vector in coordinates.

Examples

  • Standard Coordinates: Use standard basis.
    • Different coordinate systems may illuminate different properties.

Importance of Non-Standard Coordinates

  • Non-standard coordinates can make vectors and transformations more transparent.

Linear Independence and Spanning

  • Translate abstract problems to coordinate systems for computation.
  • Linear Independence: Equivalent between abstract and coordinate systems.
    • Check by row reduction.

Example: Polynomial Representation

  • Polynomials represented as vectors using standard basis.
  • Translates abstract questions into concrete coordinate questions.

Change of Basis

  • Changing basis changes coordinate system.
  • Two bases B and C translate differently.
  • Change of Basis Matrix: Translates vectors from one coordinate system to another.
    • Found via row reduction.

Computing Change of Basis Matrix

  • Use augmented matrix with basis vectors.
  • Row Reduction: Leads to identity matrix and change of basis matrix.

Example: R2 Basis Change

  • Bases: Standard (e1, e2) and Non-standard (1, 2),(3, 1).
  • Translate point V = (4, 3) in standard coordinates to non-standard coordinates.

Conclusion

  • Basis choice affects coordinate system and matrix representation.
  • Translation between coordinate systems is crucial.
  • Understanding these transformations is vital for studying linear transformations.