# Is the nonzero vector $b$ in $\mathrm{Col}\,A$ of the following $3×3$ matrix $A$

$\def\Col{\mathop{\mathrm{Col}}}\def\Nul{\mathop{\mathrm{Nul}}}$In preperation for a linear algebra exam I have coming up I was asked to construct a $3×3$ matrix $A$ and nonzero vector $b$ such that $b$ is in $\Col A$ but $b$ is not the same as any of the columns in $A$. I came up with the following:

Let $$A = \begin{bmatrix}1&-1&-1\\4&-2&-2\\1&-2&0\end{bmatrix}$$

and let $$b = \begin{bmatrix}2\\1\\1\end{bmatrix}.$$

Would this be correct? If not, what is a good answer? Any pointers someone can give on how to answer questions like this?

Can I also say $b$ is in $\Nul A$?

• To check if $b$ is in nul$A$, check if $Ab=0$. Mar 22 '18 at 4:17

So basically you have to find out nonzero vector $b$ which is in columns space of $A$ but not in columns of $A$. Take $b=pC_1+qC_2$, where $C_1$ and $C_2$ are independent columns of $A$ and $p$, $q$ are any non zero real numbers.
For your answer, consider $(2\; 1\; 1)=p(1\;4\;1)+q(-1\;-2\;-2)+r(-1\;-2\;0)$ and check that atleast two of $p,q$ and $r$ is non zero. If such $p,q$ and $r$ does not exist that it is not possible.
So what you should do is take a random column vector $v$ (say all components non-zero) and multiply with the matrix given the result will be the desired one: $b =Av$