Can it be useful to think of operations on subspaces in terms of operations in elementary set theory?

Question

This question is something that I have been thinking about on and off for a while. The basic premise is that, in Linear Algebra, a lot of the basic operations performed on subspaces have corresponding familiar operations that are performed on subsets of general sets. The operations on sets just have to be changed slightly to ensure that we always get subspaces.

Below are some examples (in each example $V$ is a vectorspace and $U,W$ are subspaces of $V$):

1. In a general set there is the concept of the empty set ($\emptyset$), which contains no elements and is a subset of every other set. However, $\emptyset$ is not a subspace. The subspace that corresponds to the $\emptyset$ would be $\{0\}$ where $0$ is the additive identity in $V$. This is clealy the smallest subspace and, like $\emptyset$, we have that $\{0\} \subseteq U$ for every subspace $U$.
2. The set $U \cap W$ is a subspaces of $V$ for all subspaces $U,W$ so this does not need to be changed.
3. $U \cup W$ may not be a subspace of $V$. The set $U + W$ can be thought of the corresponding subspace. $U \cup W$ is the smallest subset that contains both $U$ and $W$. Similarly $U + W$ is the smallest subspace that contains $U$ and $W$. We also have this relation. If $U \subseteq W$, then $U \cup W = W$ and $U + W = W$.
4. If $U$ is a subspace, then $U^c$ will never be a subspace (as it does not contain $0$). One way to think about $U^c$ is that it is the unique subset of $V$ such that $U \cup U^c = V$ and $U \cap U^c = \emptyset$. We have subspace notions of $\cup$ and $\emptyset$, so we will want the subspace $W$ that corresponds to $U^c$ to be such that $U+W=V$ and $U \cap W = \{0\}$. Clearly there could be many such subspaces $W$ but, if we are working in an inner-product space, then $W=U^\perp$ is an obvious choice.

I understant that this is quite a long post (thank you to those you have read this far). I would like to hear what people think about this, in particular:

Do you think that this is a useful\interesting way to think about subspaces and operations defined on them?

Could this way of thinking be perhaps be used as motivation when first introducing the concept of a subspace?

How misleading could this way of thinking end up being?

Thank you. Any answers/comments are greatly appreciated.