Let $q > 1$ and let $A$ be an $m \times N$ matrix with $m < N.$ Prove that there exists a $1-$sparse vector which is not a minimizer of the following optimization problem: $$\min_{\textbf{z} \in \mathbb{R}^n} \| \textbf{z} \|_{q}~\textrm{such that}~A\textbf{z}=\textbf{y}.$$
Everything in their usual notations. Any help in solving this is much appreciated.
I find the question ambiguous. You can easily find a $1$-sparse vector that does not satisfy $Az=y$. Then indeed there exists a $1$-sparse vector that is not a minimizer. The requirement $m<N$ is not necessary to prove the statement.
However, I do not think that is what the author intended to ask. This question is introduced on page 62:
I do not see why this statement is valid. The book defines $1$-sparse as vectors having at most one nonzero component (p. 3). Consider the problem $$\min_{z \in \mathbb{R}^2} \{ ||z||_q : z_1 = 0 \}$$ Note that $A = [1 \; 0]$, which is an $m \times N$ matrix with $m=1$ and $N=2$, so indeed $m<N$. The unique minimizer is $z = [0, 0]$, which is $1$-sparse.