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?


\[\text{Im }A = \text{col}(A)\]

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?


\[\\{Ax : x \in \mathbb{F}^{n\times 1}\\}\]

If $A \in \mathbb{F}^{m \times n}$, then how could you write the row space $\text{row}(A)$, as a set?


\[\\{x^\intercal A:x\in\mathbb{F}^{m\times1}\\}\]

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?


\[\begin{aligned} \text{rowrank}(A) &= \text{rowrank}(R) \\\\ &= \text{colrank}(R) \\\\ &= \text{colrank}(A) \end{aligned}\]

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?


\[T : \mathbb{F}^{n \times 1} \to \mathbb{F}^{m\times 1}\]

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 $\dim V$?


\[n\]

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 $\ker A = \ker R$?


\[\begin{aligned} Ax = 0 &\iff EAx = 0 \\\\ &\iff Rx = 0 \end{aligned}\]

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 $\text{rowrank}(A) = \text{colrank(A)}$, you get to a stage where you need to show $\text{colrank}(A) = \text{colrank}(R)$. What implies this, by the rank-nullity theorem?


\[\ker A = \ker R\]



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