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Lecture15 Projections onto Subspaces

MIT Gilberstrang 교수님의 Linear Algebra 15강 Projections onto Subspaces 강의 Projections (투영) : 하나의 vector를 다른 vector로 옮겨 표현하는 것We can see from Figure 1 that this closest point p is at the intersection formed by a line through b that is orthogonal to a. If we think of p as an approximation of b, then the length of e = b − p is the error in that approxi­mation. We could try to find p using trigonometry..

안녕하세요,오늘은 MIT Gilbert Strang 교수님의 Linear algebra 10강 four fundamental subspaces에 대해 학습하겠습니다.  Four subspacesAny m by n matrix A determines four subspaces (possibly containing only the zero vector):Column space, C(A)C(A) consists of all combinations of the columns of A and is a vector space in Rm.C(A)는 A의 열의 모든 조합으로 구성되며 Rm의 벡터 공간이다. Nullspace, N(A)This consists of all solutions x of the equation..

오늘은 MIT GilbertStrang교수님의 Linear algebra 8강에 대해 학습하겠습니다.Solvability conditions on bThe third row of A is the sum of its first and second rows, so we know that if Ax = b the third component of b equals the sum of its first and second components. If b does not satisfy b3 = b1 + b2 the system has no solution. If a combination of the rows of A gives the zero row, then the same combination of the ent..

안녕하세요 오늘은 7강 Solving Ax = 0: Pivot Variables, Special Solutions Space and Nullspace에 대해 배우겠습니다.  Solving Ax = 0: pivot variables, special solutions* Number of piviots = Rank of AThe nullspace of a matrix A is made up of the vectors x for which Ax = 0.행렬 A의 nullspace는 Ax = 0인 벡터 x로 구성된다. (중요한 점은 A행렬은 독립적이지 않다.) our algorithm for computing the nullspace of this matrix uses the method of eliminatio..

안녕하세요, 오늘은 MIT 선형대수(Linear algebra) 5강 Transposes, permutations, vector spaces에 대해 학습해보겠습니다.   Permutations(치환)Multiplication by a permutation matrix치환행렬은 행교환 (row exchange)을 수행하는 행렬이다. 행교환은 pivot이 0인 경우에 반드시 필요하다. P swaps the rows of a matrix; when applying the method of elimination we use permutation matrices to move ze­ros out of pivot positions. Our factorization A = LU then becomes PA = LU,..

안녕하세요, 오늘은 MT Gilbert Strang 교수님의 Linear Algebra 수업 Lecture4, Factorization into A = LU 에 대해 다루어보겠습니다.      Inverse of a productThe inverse of a matrix product AB is B−1 A−1. AB의 역행렬은 B−1 A−1이다.  Transpose of a product행과 열을 바꿀 때 사용the entry in row i column j of A is the entry in row j column i of A^T열I와 행J의 A를 열J와 행I으로 바꾼 것을 A^T라고 한다. The transpose of a matrix product AB is BTAT. For any invert..

안녕하세요 오늘은 MIT Gilbert Strang교수님의 Linear algebra 3강 Multiplication and Inverse Matrices를 공부하도록 하겠습니다. Matrix A multiplying 1. Column Combination행*열의 곱을 원소별 계산으로 정리할 수 있다. (대부분은 벡터로 표현)If they are square, they have got to be the same.If they are rectangular, they are not the same size.here goes A, again, times B producing C.A times a vector is a combination of the columns of A.because the columns ..

안녕하세요, 오늘은 MIT Gilbert Strang교수님의 선형대수 강의 Lecture#2 Elimination with Matrices로 공부해보도록 하겠습니다.  Method of EliminationElimination is the technique most commonly used by computer software to solve systems of linear equations. It finds a solution x to Ax = b whenever the matrix A is invertible. In the example used in class,The number 1 in the upper left corner of A is called the first pivot.The first..

안녕하세요, 인공지능 전공자라면 정말 중요하지만 어렵게 느끼는 선형대수학 입니다. 오늘은 MIT GilbertStrang 교수님의 2강인 An Overview of Linear Algebra 에 대해 리뷰해보겠습니다.   This is an overview of linear algebra given at the start of a course on the math­ ematics of engineering.  VectorsWhat do we do with vectors? Take combinationsWe can multiply vectors by scalars(such as under x1,x2,x3), add, and subtract. Given vectors u, vand w we can form ..