1 / 15

Spaces

Spaces. Various Spaces. Linear vector space: scalars and vectors Affine space adds points Euclidean spaces add distance. Scalars. Scalar field: ordinary (integer, real, complex, etc.) numbers and the operations on them -

steve
Download Presentation

Spaces

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Spaces

  2. Various Spaces • Linear vector space: scalars and vectors • Affine space adds points • Euclidean spaces add distance

  3. Scalars • Scalar field: ordinary (integer, real, complex, etc.) numbers and the operations on them - • Fundamental scalar operations: addition (+) and multiplication ( ).

  4. Scalar (II) • Associative: • Commutative: • Distributive:

  5. Scalar (III) • Additive identity (0) and multiplicative identity (1) • Additive inverse( ) and multiplicative inverse( )

  6. Vector Spaces • A vector space contains scalars and vectors • Vector addition (associative) • Zero vector

  7. Scalar-vector Multiplication • Distributive

  8. Linear Combination • Linearly independent • The greatest number of linearly independent vectors that we can find in a space gives the dimension of the space. If a vector space has dimension n, any set of n linearly independent vectors form a basis.

  9. Affine Spaces • Affine space: scalars, vectors, points • Point-point subtraction yields a vector. • Coordinate systems with/without a particular reference point:

  10. Head-to-Tail Axiom for Points

  11. Frame

  12. Euclidean Spaces • Euclidean spaces add the concept of “distance,” and thus the length of a vector. • Inner product

  13. Inner Product of Two Vectors

  14. Projections

  15. Gram-Schmidt Orthogonalization • Orthonormal basis: each vector has unit length and is orthogonal to each other

More Related