Coordinates, Dimension, and Rank

In RREF

$$\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 0\end{array}\right]$$

So, $\vec{x}$ is in the span of $B$ and $[x{]}_B=\left[\begin{array}{c}2\\ 3\end{array}\right]$.

Dimension

$\text{dim}({\mathbb{R}}^n)=n$ $\text{dim}(\text{Null}(A))=\text{no of free variables}$ $\text{dim}(\text{Col}(A))=\text{no of pivot columns}$

Rank

$\text{Rank}(A)=\text{dim}(\text{Col}(A))=\text{no of pivot columns}$ $\text{Rank}(A)+\text{dim}(\text{Null}(A))=\text{dim}(\text{Col}(A))+\text{dim}(\text{Null}(A))=\text{no of columns}$

Invertibility

If $A$ is an $n\times n$ matrix. These conditions are equivalent.

1. $A$ is invertible.
2. The columns of $A$ are a basis for ${\mathbb{R}}^n$
3. Col $A={\mathbb{R}}^n$.
4. Rank $A=\text{dim}(\text{Col }A)=n$.
5. Null $A=\{\vec{0}\}$