[G. Strang] Linear Algebra

MIT 18.06 Linear Algebra
lectured by Gilbert Strang

text: Introduction to Linear Algebra, 4th ed, Wellesley-Cambridge Press, Wellesley-Cambridge Press
official: http://math.mit.edu/linearalgebra/
google: http://books.google.co.kr/books?id=Gv4pCVyoUVYC

lecture http://web.mit.edu/18.06/www/
MIT 18.06 Linear Algebra, Spring 2005 @YouTube  videos @MIT
MIT 18.06 Linear Algebra, Spring 2010

Lecture 01: The Geometry of Linear Equations

n linear equations, n unknowns
1) Row picture -> 2 lines in x-y plane, 3 planes in x-y-z space, etc.
2) Column picture -> Linear combination of columns
3) Matrix form

Q1. Can I solve Ax=b for every b?
Q2. Do the linear combinations of the columns fill 3-D space?

"Ax is a combination of the columns of A."

Lecture 02: Elimination with Matrices

Elimination - success/failure

- augmented matrix

- pivot

Back-substitution
Elimination matrices

Matrix multiplication

- permutation matrix

- inverses

Lecture 03: Multiplication and Inverse Matrices

Matrix multiplication
Inverse of A  AB  A^t
Gauss-Jordan / Find A^-1

Lecture 04: Factorization into A=LU

Inverse of AB  A^T
Product of elimination matrices
A=LU (no row exchanges)

Lecture 05: Transpose, Permutations, Spaces R^n

section 2.7: PA = LU
section 3.1: Vector spaces and subspaces

"All the linear combinations of the columns of a matrix form a subspace, called column space C(A)."

Lecture 06: Column Space and Nullspace

Vector  spaces and subspaces
Column space of A: Solving Ax=b
Nullspace of A

Lecture 07: Solving Ax=0 : Pivot Variables, Special Solutions

Computing the nullspace (Ax=0)
Pivot Variables - free variables
Special Solutions - rref(A)=R

Lecture 08: Solving Ax = b: Row Reduced Form R

Complete solution of Ax = b
: x = x_p + x_n
Rank r
r = m : Solution exists
r = n : Soultion is unique

Lecture 09: Independence, Basis, and Dimension

Linear independence
Spanning a space
BASIS and Dimension

Lecture 10: The Four Fundamental Subspaces

(Correct an error in Lecture 9!)
Four Fundamental Subspaces (for matrix A)

Lecture 11: Matrix Spaces; Rank 1; Small World Graphs

Basis of new vector spaces
Rank one matrices
Small world graphs

http://en.wikipedia.org/wiki/Euclidean_subspace

http://en.wikipedia.org/wiki/Dimension_(vector_space)

http://en.wikipedia.org/wiki/Basis_(linear_algebra)

http://en.wikipedia.org/wiki/Rank_(linear_algebra)

Lecture 12: Graphs, Networks, Incidence Matrices

Graphs & Networks
Incidence matrices
Kirchhoff's Laws

http://en.wikipedia.org/wiki/Incidence_matrix
a matrix that shows the relationship between two classes of objects

입사 행렬(incidence matrix): 그래프 내의 모서리(edge)에 대한 정보를 담고 있는 2차원 배열. 그래프 내의 정점이 n개이면 nxn의 배열로 구성되며 배열의 각 원소는 해당 행과 열의 정점이 모서리로 연결되어 있는지를 0 또는 1로 나타낸다.  (출처 : [인터넷] 돌도끼 컴퓨터용어사전)


"Real matrices from genuine problems have structures."

"The null space tells us what combination of the columns to get zero."

"All the dependencies come from the loops."

"TREE is the name for a graph with no loop."


http://en.wikipedia.org/wiki/Planar_graph
Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely-large region), then      v − e + f = 2.

Lecture 13: Quiz 1 Review

Emphasizes Chapter 3

Lecture 14: Orthogonal vectors & subspaces

null space ㅗ row space

"Null space contains all vectors ㅗ row space."

Lecture 15: Projections onto subspace