Row Space and Null Space

Definition of row space

The row space of a matrix, $A$, denoted by $R(A)$, is the span of its row vectors. Mathematically,

$$R(A)=C(A^T)$$

Example

The row space of $A=\begin{bmatrix}1 &0&0\\0&0&1\end{bmatrix}$ is the $xz$ plane in $\R^3$, which can also be seen as a column space of $A^T$. The column space of $A$ is $\R^2$, with a basis as the first and third columns of $A$. Both the row space and column space of $A$ are two dimensional, but they’e fundamentally different.

$$rank(A)=rank(A^T)\implies dim(R(A))=dim(C(A))$$

Note: Even though $R(A)$ and $C(A)$ have the same dimensions, $R(A)$ and $C($ ...