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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 -

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Spaces

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  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

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