MIT18.06学习笔记 - Lecture 7: Solving Ax=0: pivot variables and special solutions

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

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课程提纲
1. Nullspace of $A$ and solutions
2. The Rank and Row Reduced Form

课程重点

Nullspace of $A$ and solutions

Nullspace N(A) consists of all solutions to $Ax=0$, all combinations of the special solutions.

If $A$ is invertible, $x=0$ is the only solution, otherwise there are nonzero solutions.

Computing the Nullspace
Two steps to solve $Ax=0$, $A$ is rectangular: elimination to Echelon Matrices and combinations of special solutions

The Rank and Row Reduced Form

The rank of $A$ is the number of pivots. This number is $r$.
The matrices $A$ and $U$ and $R$ have $r$ independent rows (the pivot rows) and $r$ independent columns (the pivot columns).
The rank $r$ is the dimension of the column space as well as the row space.


$Ax=uv^Tx=u(x^Tx)=0$, so $x^Tx=0$ nullspace (plane) perpendicular to row space (line):

Row Reduced Form

The nullspace matrix $N$ contains the three special solutions in it columns, so $AN=$ zero matrix:

$AX=0$ has $r$ pivots and $n-r$ free variables: $n$ columns minus $r$ pivot columns. The nullspace matrix $N$ contains the $n-r$ special solutions.
The special solutions are easy for $Rx=0$. Suppose that the first r r columns are the pivot columns: