What does Linear mean in Linear Space (Vector Space) The course I'm taking defines a vector space or Linear space as

The vector space $\mathbb{R}^n$ has a linear structure with two features: vector addition and scalar multiplication"

What does it mean by "linear structure" here? (and what makes it linear)
 A: Let's zoom out a bit.
In math, there are these things called algebraic structures. Their role is similar to the role of interfaces in Java programming, or typeclasses in Haskell programming.
If you ask a Java programmer what a list is, they might reply,

In Java, the formal definition of a list is given by the List interface. For an object to qualify as a List, you have to be able to add elements to it, retrieve elements from it, search through it, ask whether or not it's empty, and do all the other stuff the interface definition says you should be able to do with a List.

If you ask a Haskell programmer what a number is, they might reply,

In Haskell, the formal definition of a number is given by the Num typeclass. For a bunch of data to qualify as Num data, you have to be able to add them together, subtract them from each other, multiply them, negate them, and do all the other stuff the typeclass definition says you should be able to do with Num data.

If you ask a mathematician what a symmetry is, they might reply,

In math, the formal definition of a symmetry is given by the Group structure. For a bunch of actions to qualify as a Group, you have to be able to reverse them, perform one of them after the other, find one of them that has no effect, and do all the other stuff the structure definition says you should be able to do with Group elements.

In summary: algebraic structures, like interfaces and typeclasses, are a way of formally defining how a certain kind of thing or a certain bunch of things ought to act.

A linear structure, more commonly known these days as a vector space structure, is a kind of algebraic structure. Roughly speaking, a vector space structure formally defines how things that "act like arrows drawn on a page" are supposed to behave. For a bunch of things to qualify as a vector space, you have to be able to:


*

*Reverse them, making them point the other way.

*Scale them, making them longer or shorter.

*Scale them all the way down to zero length.

*Place two of them "tip to tail" to create a new one.


My descriptions of these operations are designed to make you think about manipulating arrows drawn on a page. However, they can be defined in very different ways, just as the add method of a Java List can be defined to do something very different than what you might expect. Fortunately, there's a longish list of rules the operations in a vector space have to follow, which ensure thaty they act at least a little bit the familiar operations on arrows.
A: Other people alluded to the formal definition about being closed under linear combinations. Perhaps what you're looking for is a slightly more intuitive, less rigorous answer.
A finite dimensional vector space is isomorphic to some $\mathbf{R}^n$ for some $n$. So for instance, a $1$-dimensional vector space "looks like" the real number line, and a $2$-dimensional vector space "looks like" a plane.
Continuing with this reasoning, I offer the following answer to your question.

A "linear" space is a space which is "flat". So linear essentially means flat in this context.

