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.