Notes - Linear Algebra I MT22, Column rank and row rank
Flashcards
If $A$ is a matrix, how can you write $\text{Im }A$ 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 $\text{col}(A)$, or equivalently the image $\text{Im} A$, as a set?
If $A \in \mathbb{F}^{m \times n}$, then how could you write the row space $\text{row}(A)$, 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 $\text{rowrank}(A) = \text{colrank(A)}$, 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 $\text{Im } A$, 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 $\dim(\ker A) + \dim(\text{Im } A)$, what is $\dim V$?
Let $A \in \mathbb{F}^{m \times n}$, and its RREF $R \in \mathbb{F}^{m \times n}$. Can you give a quick proof that $\ker A = \ker R$?
as $E$ is invertible (representing the EROs).