If W is a subspace, is it also a linear space? If V is a linear space, is it also a subspace? I am having trouble wrapping my head around the difference between the two, as it seems that the way the book defines them is the following: both have to have a zero (neutral) element, both are closed under addition and scalar multiplication. Thanks!
[Math] the difference between linear space and a subspace
linear algebra
Best Answer
The main difference between refering to a vector spaces as a linear space or as a subspace is, unsurprisingly, context.
When one talks about a "subspace", one is thinking of it as being "inside" another vector space, with the inherited operations. So, if we refer to the set of all polynomials of degree at most $2$ with coefficients in $\mathbb{R}$ as a "subspace of the vector space of all polynomials" (note that subspace is relation term; something is a subspace of something else, thought the "something else" may be left implied or understood from context), then we are not merely thinking of this set as a vector space, we are thinking of it as a vector space sitting inside a larger vector space. On the other hand, if we refer to the set as "the vector space of all polynomials of degree at most $2$ with coefficients in $\mathbb{R}$", then we are not really interested in the fact that it is sitting inside a larger vector space, but we want to consider it as a vector space in and of itself.
We do that in set theory all the time, since a set may be a set or a subset of something else. The natural numbers form a set, and we refer to "the set of natural numbers" when we are focusing on the natural numbers exclusively; on the other hand, sometimes we want to think about the natural numbers and their relationship with the larger set of rationals or real numbers, so we can talk about "the subset of the real numbers consisting of the natural numbers."
Now, every vector space is a subspace of itself; every subspace of a vector space is itself a vector space (with the inherited operations). So whether we call them "space" or "subspace of
<something>" is entirely a question of whether we want to focus on the object itself, or consider it in context with something else.