  Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces
$\ell_M^p$ associated with it. When equipped with a suitable norm
$\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of
sequences $(\ell_M^p,\|\cdot\|_{M,p})$. In particular we show that such spaces
are separable and strictly/uniformly convex for a considerably large class of
infinite matrices $M$ for all $p>1$. A special attention is given to the
identification of the dual space $(\ell_M^p )^*$. Building on the earlier works
of Bennett and J\"agers, we extend and apply some classical factorization
results to the sequence spaces $\ell_M^p$.
