Chapter 3.2, Problem 57E

Consider the matrix $A=\left[\begin{array}{ccccccc}0& 1& 2& 0& 0& 3& 0\\ 0& 0& 0& 1& 0& 4& 0\\ 0& 0& 0& 0& 1& 5& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right].$

Note that matrix A is in reduced row-echelon form.

For which positive integers $j=1,\mathrm{...7}$ doesthere exist a vector $\stackrel{\to }{x}$ in the kernel of A such that thejth component $x_j$ of $\stackrel{\to }{x}$ is nonzero, while all the components $x_j+1,\mathrm{...},x_7$ are zero?