To define a new sequence space and determine the Köthe-Toeplitz duals of this sequence space, characterizing the matrix transformation classes between the defined sequence spaces and classical sequence spaces has been an important area of work for researchers. Defining and examining a new vector-valued sequence space is also a considerable field of study since it generalizes classical sequence spaces. In this study, new vector-valued sequence spaces $E(X,\lambda ,p)$ are introduced. The Köthe-Toeplitz duals of $E(X,\lambda ,p)$ spaces are identified. Also, necessary and sufficient conditions are determined for $A=(A_{nk})$ to belong to the matrix classes $(E(X,\lambda ,p),c(q))$; where $A_{nk}\in B(X,Y)$, $X\in \{c,{\ell }_{\mathrm{\infty }}\}$ and $Y$ is any Banach spaces.