This question really only concerns terminology. In the linear algebra lectures that I am watching, the professor refers to the "linear structure" of a vector space. I know the definition of linearity in the context of a linear transformation, but that's a map between vector spaces. The vector spaces themselves do not seem to have "linear structure." I cannot figure out what exactly this term means. I believe another name for vector space is a "linear space," and that this must be related, but I cannot figure out what this could be referring to unless the ability to take linear combinations is the goal.
Vector space and “linear structure”
vector-spaces
Best Answer
As you know, a vector space $V$ over a field $\mathbb{F}$ is endowed with two different "structures", one is given by addition, $+$, and gives the set $V$ a structure of abelian group $(V, +)$, the other one is given by multiplication of vectors in $V$ by elements in $F$, called scalars, and satisfies some axioms. The result is that the set $V$ of vectors must be closed with respect to both addition and scalar multiplication. That is, if $v$ and $v'$ are vectors in $V$, then also the sum $v+v'$ must be a vector in $V$ and if $\lambda\in\mathbb{F}$ is a scalar, then $\lambda v$ must be a vector in $V$. You may summarize these closure properties by saying that $V$ is closed under linear combinations of vectors, that is $\lambda v+\lambda'v'$ is in $V$ for all $\lambda,\lambda'\in\mathbb{F}$ and $v,v'\in V$. Vectors of the form $\lambda v+\lambda'v'$ are called linear combinations of $v$ and $v'$. This can be a reason for calling a vector space a linear space, because it is closed under linear combinations of vectors. Linear combinations appear everywhere in vector spaces: if you fix a basis for the space, then every vector can be written (uniquely) as a linear combination of vectors in the basis.