Linear Algebra I MT22, Column rank and row rank
Flashcards
If $A$ is a matrix, how can you write $\mathrm{Im}\,A$ in terms of the column space or row space?
If $A \in \mathbb{F}^{m\times n}$, then how could you write the column space $\mathrm{col}(A)$, or equivalently the image $\mathrm{Im} A$, as a set?
If $A \in \mathbb{F}^{m \times n}$, then how could you write the row space $\mathrm{row}(A)$, as a set?
Let $A \in \mathbb{F}^{m \times n}$, and its RREF $R \in \mathbb{F}^{m \times n}$. When proving that the $\mathrm{rowrank}(A) = \mathrm{colrank}(A)$, what long line of equalities do you make?
Let $A \in \mathbb{F}^{m \times n}$. When talking about the kernel or image of a matrix, what linear transformation $T$ are we referring to?
Let $A \in \mathbb{F}^{m \times n}$. When talking about $\ker A$ or $\mathrm{Im}\, A$, we are really referring to the kernel and image of a linear transformation $T$:
\[T : \mathbb{F}^{n \times 1} \to \mathbb{F}^{m\times 1} \]
Therefore, when we apply the rank-nullity theorem to $\dim(\ker A) + \dim(\mathrm{Im}\, A)$, what is $\dim V$?
Let $A \in \mathbb{F}^{m \times n}$, and its RREF $R \in \mathbb{F}^{m \times n}$. @Prove that $\ker A = \ker R$.
as $E$ is invertible (representing the EROs).
Let $A \in \mathbb{F}^{m \times n}$, and its RREF $R \in \mathbb{F}^{m \times n}$. When proving that the $\mathrm{rowrank}(A) = \mathrm{colrank}(A)$, you get to a stage where you need to show $\mathrm{colrank}(A) = \mathrm{colrank}(R)$. What implies this, by the rank-nullity theorem?
Bite-sized
Row rank equals column rank: for any $A \in \mathbb F^{m \times n}$, $\mathrm{rowrank}(A) = $ $\mathrm{colrank}(A)$, so the unqualified “rank” is well-defined. In NLA this is just $\mathrm{rank}(A) = $ the number of nonzero singular values of $A$ — the SVD makes the row/column duality manifest.
What does $\mathrm{rank}(A) = r$ tell you about the SVD of $A$?
$A$ has exactly $r$ nonzero singular values, so $A = U _ r \Sigma _ r V _ r^\top$ in “compact” SVD form where $U _ r \in \mathbb R^{m \times r}$, $\Sigma _ r \in \mathbb R^{r \times r}$ with positive diagonal, and $V _ r \in \mathbb R^{n \times r}$. The first $r$ columns of $U$ span the column space ($= \mathrm{Im}\,A$), the first $r$ columns of $V$ span the row space, and the last $n - r$ columns of $V$ span $\ker A$.