TITLE: Please tell me if I'm right
QUESTION [0 upvotes]: I'm reading about vector spaces and I'm too confused with the terms. Please let me know if the following inferences are correct.

If I perform a linear operation on the components of a vector, then the result belongs to a dual space.
If I perform a linear operation on vectors, then the result belongs to a vector space.

REPLY [1 votes]: for $2$:
yes. If not, we have a linearity property, which says roughly that for a linear transformation $f:V \to W$, $T(v+u)=T(v)+T(u)$. But note that $T(v), T(u) \in W$, so $W$ should be a space where addition makes sense, and $T(kv)=k T(v)$, for some scalar $k$, so scalar multiplication should also make sense. This essentially means that to define a map as linear, kind of requires that after applying a linear operation, the result belongs to a vector space.
for $1$: It is unclear what you mean to me, but I would say that the dual space consists of  linear operations you can do on a vector, so that the result is a scalar. In particular, it consists of linear operations $\phi:V \to k$, where $k$ is the base field.
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