[Math] Determining if a vector is in the row space

linear algebravector-spaces

I am just trying to determine if the vector $[0, 7, 4]$ belongs in the row space of the matrix $$A = \begin{bmatrix}
1 & 2 & 0 \\
3 & -1 & 4 \\
1 & -5 & 4 \\
\end{bmatrix}
$$
What I have done so far is created an augmented matrix like so
$$ \left[
\begin{array}{ccc|c}
1 & 3 & 1 & 0 \\
2 & -1 & -5 & 7 \\
0 & 4 & 4 & 4 \\
\end{array}
\right]
$$
(naive gaussian) reducing to
$$ \left[
\begin{array}{ccc|c}
1 & 3 & 1 & 0 \\
0 & -7 & -7 & 7 \\
0 & 0 & 0 & 32/7 \\
\end{array}
\right]
$$
returning inconsistent, i.e. not existing in the row space. However apparently it does in fact belong in the row space, so clearly I have gone about this the wrong way. Is someone able to correct this?

Best Answer

A vector $\vec b$ is in the row space of $A$ if and only if $\vec b$ is in the column space of $A^\top$. Thus, to determine if the vector $\vec b=\left[\begin{array}{r} 0 \\ 7 \\ 4 \end{array}\right]$ is in the row space of $A = \left[\begin{array}{rrr} 1 & 2 & 0 \\ 3 & -1 & 4 \\ 1 & -5 & 4 \end{array}\right]$, form the augmented matrix $$ \left[\begin{array}{rrr|r} 1 & 3 & 1 & 0 \\ 2 & -1 & -5 & 7 \\ 0 & 4 & 4 & 4 \end{array}\right] $$ Row reducing gives $$ \DeclareMathOperator{rref}{rref}\rref\left[\begin{array}{rrr|r} 1 & 3 & 1 & 0 \\ 2 & -1 & -5 & 7 \\ 0 & 4 & 4 & 4 \end{array}\right]= \left[\begin{array}{rrr|r} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] $$ This gives an inconsistent system. Hence $\vec b$ is not in the row space of $A$.

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