# Construct a 3\times3 matrix A, with nonzero entries, and a vector b in

Construct a $3×3$ matrix A, with nonzero entries, and a vector b in ${\mathbb{R}}^{3}$ such that b is not in the set spanned by the columns of A.
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Sally Cresswell

For the matrix A in the result, any vector in the span of the column vectors is of the form on the left.
${x}_{1}\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]+{x}_{2}\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]+{x}_{3}\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}{x}_{1}+{x}_{2}+{x}_{3}\\ {x}_{1}+{x}_{2}+{x}_{3}\\ {x}_{1}+{x}_{2}+{x}_{3}\end{array}\right]$
After multiply the scalars and adding the vectors, we get a vector of this form. Notice that all the entries vector are equal. Since the entries in vector b are not all same, 1,2,3,b cannot be in the span of the column vectors.
Result:
$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$
and
$b=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$