I was reading a linear algebra book when I came across a statement that confused me. Help me if I misunderstood anything.

Question

A book that I'm reading stated "let $V$ be a vector space and let $D$ be a nonempty subset of $V$. Let $M$ be the collection of all vectors in $V$ which can be expressed as a linear combination of some finite subset of $D$. Then $M$ is a subspace spanned by $D$." To my understanding(I believe the confusion is here) $D$ should be collection of independent vectors. What I understood was that any subset of the null space of a vector space can be expressed as the linear combination of finite subset of the null space that are linearly independent to one another. For example if $X^1=(1,1,0)$ and $X^2=(0,1,1)$ then a subset of the null space $X$ can be expressed as $X=aX^1+bX^2$ where $a$ & $b$ are real numbers. But then the book continues to prove that $M$ is a subspace by concluding $0$ is a linear combination of finite subset of $D$ but last time I read a set containing the zero vector can't be linearly independent. I'm pretty sure that I have misunderstood a concept, I just dont know what mistake I have made.