Does adding two vectors from different subspaces have meaning? Given two vectors $\langle1,1\rangle$ and $\langle2,2,2\rangle$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively, is it possible to add them together?
If we add them in order, increasing the resulting dimension to $\mathbb{R}^5$, we get $
\langle1,1,2,2,2\rangle$. However, this is not a commutative result as $\langle1,1\rangle + \langle2,2,2\rangle = \langle1,1,2,2,2\rangle$ would not be equal to $\langle2,2,2\rangle+ \langle1,1\rangle =\langle2,2,2,1,1\rangle$.
Additionally, adding the vectors component wise seems illogical as $\langle1,1\rangle + \langle2,2,2\rangle = \langle1+2,1+2, \text{unknown}+ 2\rangle$. And assuming the unknown to be $0$ seems arbitrary.
I have a gut reaction, that this operation shouldn't be undefined, but I am new to linear algebra and don't have a strong conceptualization of it yet. (This is my first post. Yay!!)
 A: You've proposed two kinds of operations here. The first corresponds to what is commonly called 'lifting,' while the second is certainly illegal (techincally, the first is called concatenation, but that is literally just conjoining ordered values, and doesn't have the kind of linear algebraic interest as lifting).
To explain why the first is called, lifting, we turn to $\mathbb{R}^2$ and $\mathbb{R}^3$ for intuition.
Consider a vector $v \in \mathbb{R}^2$. It is natural to consider this vector on the $z = 0$ plane in $\mathbb{R}^3$, so when we form the vector:
$$\begin{bmatrix} v \\ 1 \end{bmatrix}$$
we are 'lifting' the plane up one unit. It's certainly not like adding, and has none of the nice properties that addition has. That said, it is a useful tool used to derive identities that show certain things to be true; the idea is that if we take some point arrangement and want to prove something about it (often in convex geometry), it might be useful to lift up the arrangement which keeps its properties, but forces information on the values which allows us to retain some of that information when we 'bring it back down' to its original dimension.
The second is ill-defined, unless you stipulate what value fills the missing spots. It is true that such a thing is arbitrary, but it may be useful to embed the lower-dimensional vector in the higher dimensional space, in which case you can add them in a way that is analogous to the second way.
