Learning Reflection on Linear Algebra (Mostly about dimensions)

For a $m\times n$ matrix, it has \rank A\le m

If we need to know the minimum of $\dim \text{Nul}\;A$, it is same to know the maximum of \rank A.

Row Operation will change $\text{Col}\; A$ , but not \rank A

Let $E$ be an invertible matrix:

• $\text{Col}(EA)=E\,\text{Col}(A)$ : just like the row operations
• $\text{Col}\;(AE)=\text{Col}(A)$ : only a new linear combinations of the columns of $A$

\rank(\text{Row}\,A)=\rank(\text{Col}\,A)

Consider the REF of $A$, then :

• \rank(\text{Row}\,A) : the number of nonzero rows
• \rank(\text{Col}\,A) : the number of pivot positions

Let $A$ be a $m\times n$ matrix, then:

• $\text{Nul}\,A\subseteq \R^n$
• $\text{Col}\,A\subseteq \R^m$

For The Invertible Theorem, we have:

If $A$ is invertible, then $Ax=b$ has at least one solution.

But, in fact, it has only one solution.

If $x\in \text{Col}\,A$, then: $\exist y\;\Rightarrow\; Ay=x$

If $\exist y\;\Rightarrow\; Ay=x$, then $x\in \text{Col}\,A$

Prove unique:

1. Prove existence: by definition
2. Prove unique : Use contradiction. First assume it has at least two…