From 3ccb2bc0c514465bc98058fb4ec2e1d169ed36cd Mon Sep 17 00:00:00 2001 From: Shervine Amidi Date: Sat, 14 Nov 2020 14:40:12 -0800 Subject: [PATCH] Delete refresher-linear-algebra.md --- ko/refresher-linear-algebra.md | 339 --------------------------------- 1 file changed, 339 deletions(-) delete mode 100644 ko/refresher-linear-algebra.md diff --git a/ko/refresher-linear-algebra.md b/ko/refresher-linear-algebra.md deleted file mode 100644 index a6b440d1e..000000000 --- a/ko/refresher-linear-algebra.md +++ /dev/null @@ -1,339 +0,0 @@ -**1. Linear Algebra and Calculus refresher** - -⟶ - -
- -**2. General notations** - -⟶ - -
- -**3. Definitions** - -⟶ - -
- -**4. Vector ― We note x∈Rn a vector with n entries, where xi∈R is the ith entry:** - -⟶ - -
- -**5. Matrix ― We note A∈Rm×n a matrix with m rows and n columns, where Ai,j∈R is the entry located in the ith row and jth column:** - -⟶ - -
- -**6. Remark: the vector x defined above can be viewed as a n×1 matrix and is more particularly called a column-vector.** - -⟶ - -
- -**7. Main matrices** - -⟶ - -
- -**8. Identity matrix ― The identity matrix I∈Rn×n is a square matrix with ones in its diagonal and zero everywhere else:** - -⟶ - -
- -**9. Remark: for all matrices A∈Rn×n, we have A×I=I×A=A.** - -⟶ - -
- -**10. Diagonal matrix ― A diagonal matrix D∈Rn×n is a square matrix with nonzero values in its diagonal and zero everywhere else:** - -⟶ - -
- -**11. Remark: we also note D as diag(d1,...,dn).** - -⟶ - -
- -**12. Matrix operations** - -⟶ - -
- -**13. Multiplication** - -⟶ - -
- -**14. Vector-vector ― There are two types of vector-vector products:** - -⟶ - -
- -**15. inner product: for x,y∈Rn, we have:** - -⟶ - -
- -**16. outer product: for x∈Rm,y∈Rn, we have:** - -⟶ - -
- -**17. Matrix-vector ― The product of matrix A∈Rm×n and vector x∈Rn is a vector of size Rn, such that:** - -⟶ - -
- -**18. where aTr,i are the vector rows and ac,j are the vector columns of A, and xi are the entries of x.** - -⟶ - -
- -**19. Matrix-matrix ― The product of matrices A∈Rm×n and B∈Rn×p is a matrix of size Rn×p, such that:** - -⟶ - -
- -**20. where aTr,i,bTr,i are the vector rows and ac,j,bc,j are the vector columns of A and B respectively** - -⟶ - -
- -**21. Other operations** - -⟶ - -
- -**22. Transpose ― The transpose of a matrix A∈Rm×n, noted AT, is such that its entries are flipped:** - -⟶ - -
- -**23. Remark: for matrices A,B, we have (AB)T=BTAT** - -⟶ - -
- -**24. Inverse ― The inverse of an invertible square matrix A is noted A−1 and is the only matrix such that:** - -⟶ - -
- -**25. Remark: not all square matrices are invertible. Also, for matrices A,B, we have (AB)−1=B−1A−1** - -⟶ - -
- -**26. Trace ― The trace of a square matrix A, noted tr(A), is the sum of its diagonal entries:** - -⟶ - -
- -**27. Remark: for matrices A,B, we have tr(AT)=tr(A) and tr(AB)=tr(BA)** - -⟶ - -
- -**28. Determinant ― The determinant of a square matrix A∈Rn×n, noted |A| or det(A) is expressed recursively in terms of A∖i,∖j, which is the matrix A without its ith row and jth column, as follows:** - -⟶ - -
- -**29. Remark: A is invertible if and only if |A|≠0. Also, |AB|=|A||B| and |AT|=|A|.** - -⟶ - -
- -**30. Matrix properties** - -⟶ - -
- -**31. Definitions** - -⟶ - -
- -**32. Symmetric decomposition ― A given matrix A can be expressed in terms of its symmetric and antisymmetric parts as follows:** - -⟶ - -
- -**33. [Symmetric, Antisymmetric]** - -⟶ - -
- -**34. Norm ― A norm is a function N:V⟶[0,+∞[ where V is a vector space, and such that for all x,y∈V, we have:** - -⟶ - -
- -**35. N(ax)=|a|N(x) for a scalar** - -⟶ - -
- -**36. if N(x)=0, then x=0** - -⟶ - -
- -**37. For x∈V, the most commonly used norms are summed up in the table below:** - -⟶ - -
- -**38. [Norm, Notation, Definition, Use case]** - -⟶ - -
- -**39. Linearly dependence ― A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others.** - -⟶ - -
- -**40. Remark: if no vector can be written this way, then the vectors are said to be linearly independent** - -⟶ - -
- -**41. Matrix rank ― The rank of a given matrix A is noted rank(A) and is the dimension of the vector space generated by its columns. This is equivalent to the maximum number of linearly independent columns of A.** - -⟶ - -
- -**42. Positive semi-definite matrix ― A matrix A∈Rn×n is positive semi-definite (PSD) and is noted A⪰0 if we have:** - -⟶ - -
- -**43. Remark: similarly, a matrix A is said to be positive definite, and is noted A≻0, if it is a PSD matrix which satisfies for all non-zero vector x, xTAx>0.** - -⟶ - -
- -**44. Eigenvalue, eigenvector ― Given a matrix A∈Rn×n, λ is said to be an eigenvalue of A if there exists a vector z∈Rn∖{0}, called eigenvector, such that we have:** - -⟶ - -
- -**45. Spectral theorem ― Let A∈Rn×n. If A is symmetric, then A is diagonalizable by a real orthogonal matrix U∈Rn×n. By noting Λ=diag(λ1,...,λn), we have:** - -⟶ - -
- -**46. diagonal** - -⟶ - -
- -**47. Singular-value decomposition ― For a given matrix A of dimensions m×n, the singular-value decomposition (SVD) is a factorization technique that guarantees the existence of U m×m unitary, Σ m×n diagonal and V n×n unitary matrices, such that:** - -⟶ - -
- -**48. Matrix calculus** - -⟶ - -
- -**49. Gradient ― Let f:Rm×n→R be a function and A∈Rm×n be a matrix. The gradient of f with respect to A is a m×n matrix, noted ∇Af(A), such that:** - -⟶ - -
- -**50. Remark: the gradient of f is only defined when f is a function that returns a scalar.** - -⟶ - -
- -**51. Hessian ― Let f:Rn→R be a function and x∈Rn be a vector. The hessian of f with respect to x is a n×n symmetric matrix, noted ∇2xf(x), such that:** - -⟶ - -
- -**52. Remark: the hessian of f is only defined when f is a function that returns a scalar** - -⟶ - -
- -**53. Gradient operations ― For matrices A,B,C, the following gradient properties are worth having in mind:** - -⟶ - -
- -**54. [General notations, Definitions, Main matrices]** - -⟶ - -
- -**55. [Matrix operations, Multiplication, Other operations]** - -⟶ - -
- -**56. [Matrix properties, Norm, Eigenvalue/Eigenvector, Singular-value decomposition]** - -⟶ - -
- -**57. [Matrix calculus, Gradient, Hessian, Operations]** - -⟶