linear algebra-question on a linear transformation4 Let V be a vector space and W be any subspace. Then for given a vector space Y can you give a linear transformation such that W is the kernel of it?

you need a constraint the dimension of $Y$ because of $dim(ker(A) + dim(image(A) = dim(V).$ So if $dim(Y) = m \ge dim(image(A) \ge dim(V)= n - dim(W) = k.$ 
Here are the steps to construct  $A:$
(a) Find a basis $\{e1, e2, \ldots , ek\}$ for $W.$
(b) Extend the basis to $\{e1, e2, \ldots, en  \}$
(c) Find a basis $\{ f1, f2, \ldots fm\}$ of $Y.$
(d) Set $Ae1 = Ae2 = \ldots = 0,$ Aek+1 = f1, \ldots, Aen = fn-k.$
(e) $A(x1e1 + \ldots xnen) = x1 Ae1 + \dots xn Aen.$
