MIT18.06學習筆記 - Lecture 7: Solving Ax=0: pivot variables and special solutions

這個系列文章是我重溫Gilbert老爺子的線性代數在線課程的學習筆記。
Course Name：MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章節內容: 3.2-3.3

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課程提綱
1. Nullspace of A$A$ and solutions
2. The Rank and Row Reduced Form

課程重點

Nullspace of A$A$ and solutions

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

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

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

The Rank and Row Reduced Form

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


Ax=uvTx=u(xTx)=0$Ax=u{v}^{T}x=u(x^{T}x)=0$ , so xTx=0$x^{T}x=0$ nullspace (plane) perpendicular to row space (line):

Row Reduced Form

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

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