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 - Fundamental scalar operations: addition (+) and multiplication ( ).

Scalar (II) Associative: Commutative: Distributive:

Scalar (III) Additive identity (0) and multiplicative identity (1) Additive inverse( ) and multiplicative inverse( )

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

Scalar-vector Multiplication Distributive

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.

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

Head-to-Tail Axiom for Points

Frame

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

Inner Product of Two Vectors

Projections

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