Exercise 1.1.14

Question

Suppose two matrices $A$ and $B$ have the same column space.
\begin{enumerate}[label=(\alph*)]
\item Show that their row spaces can be different.
\item Show that the matrices $C$ (basic columns) can be different.
\item What number will be the same for $A$ and $B$?
\end{enumerate}

Neil Z. · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item
    We just need to give an example of two matrices $A$ and $B$ that
    have the same column space but different row spaces. Consider
    vectors in $R^2$, let
    $A=\begin{bmatrix}1 &2 \ 1 & 2\end{bmatrix}$, and
    $B=\begin{bmatrix}2 &1 \ 2 & 1\end{bmatrix}$. Their column spaces
    are the same, i.e.~the line $x\begin{bmatrix}1 \ 1\end{bmatrix}$.\par
If we write them in $A=CR$ format, we have
$A=\begin{bmatrix}1\1\end{bmatrix}\begin{bmatrix}1 & 2 \end{bmatrix}$
and
$B =\begin{bmatrix}1\1\end{bmatrix}\begin{bmatrix}2 & 1 \end{bmatrix}$,
we can see that the row space of $A$ is line
$x\begin{bmatrix}1 & 2 \end{bmatrix}$, whic is different from the row
space of $B$ is line $\begin{bmatrix}2 & 1 \end{bmatrix}$.

\item
    The matrix $C$ for $A$ is $\begin{bmatrix}1\1\end{bmatrix}$,
    while it is $\begin{bmatrix}2\2\end{bmatrix}$ for $B$.

\item
    $A$ and $B$ have the same rank because their column spaces are
    the same.
\end{enumerate}

Exercise 1.1.14

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Exercise 1.1.14

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