1. Vector Space Examples Definition of vector space
x-y space：R2，
x-y-z space：R3
Definition of vector space
Closure for addition and scalar multiplication
Addition：commutative law，associative law，zeros， inverse
Scalar multiplication： associative law，associative law for vector and scalar，unity law
Other examples for vector space?

2. Vector Space Vector space properties Real vector space
Addition：unique zero vector，unique inverse
Scalar multiplication：
Addition inverse：
Real vector space
Complex vector space

3. Vector Space Definition of vector space
Closure for addition and scalar multiplication
Addition：commutative law，associative law，zeros， inverse
Scalar multiplication： associative law，associative law for vector and scalar，unity law

4. Subspace Definition：subset + space Operation of subspace：
Subspace addition：
Subspace intersection：
Subspace addition and intersection, both subspace
Subspace addition，the smallest subspace including A and B
Subspace intersection，the largest space included in A and B

5. Subspace Operations
Subspace addition，the smallest subspace including A and B

6. Linear Transform Linear transform Linear transform properties
Additivity：
Homogeneity：
Linear transform properties
Map: Linear subspace to linear subspace
Map: Domain of linear subspace to linear subspace
Zero subspace of linear transform?
Representation of linear transform?

7. Linear Transform Representation of Linear Transform
Basis of domain subspace:
Mapping of domain basis:
Range of linear transform:
Zero subspace of linear transform?

8. Inner Product，Inner Product Subspace
Conjugate：
Linear：
Non-negativity：
Inner product subspace：
The subspace where the inner product has been defined

9. Norm，Normed Vector Subspace
Vector norm：
Non-negativity：
Homogeneity ：
Triangular inequality：
Normed vector space
Cauchy sequence：
Banach space：Cauchy sequence converges
Hilbert space： Cauchy sequence converges in terms of norm

10. Vector Inner Product Vector inner product definition
Weighted inner product
Function inner product
Function weighted inner product

11. Properties Polarization identity, real space
Polarization identity, complex space
Parallelogram law
Triangular law
Cauchy-Schwartz inequality

12. Proof Outline Polarization identity, real space
Polarization identity, complex space
Parallelogram law
Triangular law
Cauchy-Schwartz inequality

13. Typical Vector Norm L0 Norm： number of nonzero elements L1 Norm：
Lp Norm：

14. Typical Vector Norm
Lp norm versus the Lp norm

15. Matrix Inner Product and Norm
Matrix inner product properties
Non-negativity：
Scalar multiplication:
Triangular inequality:
Norm of multiplication:
Matrix norm examples

16. Induced Norm, p-Norm Definition of induced p-norm
Example on the p-norm

17. Induced Norm, p-Norm
Proof of p-Norm

18. Entry-wise Norm Entry-wise norm definition
Examples on the entry-wise norm

19. Properties Properties on the matrix norm
Properties on the inner product and norm