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