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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.
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