Solve Unique Decomposition Problem: Linear Algebra Vectors & Scalars

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Let V be a vector space and ℓ : V → R be a linear map. If z ∈ V is not in the
nullspace of ℓ, show that every x ∈ V can be decomposed uniquely as x = v + cz ,
where v is in the nullspace of ℓ and c is a scalar.
 
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I'll give the start of the proof:

There are two situations:
1) x is in the nullspace of l, then the statement is trivial.
2) x is not in the nullspace of l, then l(x)\neq 0 and l(z)\neq 0. Thus, there exists a nonzero c, such that l(x)=cl(z). Try to continue the argument (hint: what happens to x-cz?
 

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