MIT18.06学习笔记 - Lecture 10: The four fundamental subspaces

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: 3.6

---

课程提纲
1. Four fundamental subspaces for matrix $A$

课程重点

Four fundamental subspaces for matrix $A$

The rank of a matrix is the number of pivots. The dimension of a subspace is the number of vectors in a basis.

The main point is that the four dimensions are the same for $A$ and $R$.
The dimension of the row space is the rank $r$. The nonzero rows of $R$ form a basis.
The dimension of the column space is the rank $r$. The pivot columns form a basis.
The nullspace has dimension $n-r$. The special solutions form a basis.
The nullspace of $R^T$ (left nullspace of $R$) has dimension $m-r$.

The big picture