这个系列文章是我重温Gilbert老爷子的线性代数在线课程的学习笔记。
Course Name:MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章节内容: 3.2-3.3
课程提纲
1. Nullspace of and solutions
2. The Rank and Row Reduced Form
课程重点
Nullspace of and solutions
Nullspace N(A) consists of all solutions to , all combinations of the special solutions.
If is invertible, is the only solution, otherwise there are nonzero solutions.
Computing the Nullspace
Two steps to solve , is rectangular: elimination to Echelon Matrices and combinations of special solutions
The Rank and Row Reduced Form
The rank of is the number of pivots. This number is .
The matrices and and have independent rows (the pivot rows) and independent columns (the pivot columns).
The rank is the dimension of the column space as well as the row space.
, so nullspace (plane) perpendicular to row space (line):
Row Reduced Form
The nullspace matrix contains the three special solutions in it columns, so zero matrix:
has pivots and free variables: columns minus pivot columns. The nullspace matrix contains the special solutions.
The special solutions are easy for . Suppose that the first columns are the pivot columns: