In Guillemin and Haine’s text Differential Forms, they define a redundant k-tensor as a so called decomposable tensor $I$: $I=l_{1} \otimes \cdots \otimes l_{k}$, where each $l_{i}$ is a dual vector of a vector space $V$. The authors go on to construct the exterior vector space as the quotient space: $$\Lambda ^k (V^{*}) = L^k (V)/ I^k (V)$$ where $L^k (V)$ is the set of all k-tensors over $V$ and $I^k (V)$ is the set of the linear span all redundant k-tensors.
My question is is $I^k(V)$ a vector space? Because if one tensor is redundant in one slot, and another redundant tensor is redundant in another slot, then their sum will not necessarily be redundant in either of the slots.