A question reads: consider the vector w = (2,3,9,1). Which (if any) of the following vectors: ... are in the same translate of V as w?
And the ellipsis is filling in for three other vectors.
This question is part b of a bigger question, the first part of which involved finding the matrix of a linear transformation (whose domain is R4 and codomain R2) given it's kernel (which is V). I'm also told in the question that V is a subspace of R4 and is spanned by two vectors (whose four elements are given).
I'm guessing this involves the transformation's matrix in some way, but I'm not sure on what "the same translate" means here. Could anyone offer some advice?
I would write the translate of a subspace $V$ as $x + V$ where $x$ is a vector. This is shorthand for the set notation $\{x + v \mid v \in V\}.$ Note that $x$ itself is always a member of the translate $x + V,$ if $x$ is a member of a certain translate of $V$ then one of the ways to name that translate is $x + V.$
So you're looking for a vector that is a member of $w + V,$ that is, if the vector is named $u,$ then $u = w + v$ where $v \in V.$ Now consider what happens when you apply the given transformation $L$ to that vector: $L(w + v) = L(w) + L(v).$ You should be able to simplify that further.
Even better, look what you get if you do $L(u - w).$