We sometimes right a column vector as $(x_1,x_2,x_3)$ with parentheses (to save space and not confuse it with a row vector), we also tend to write euclidean vectors in this way, in both cases it looks like a 'tuple', we know that $\mathbb R^n$ is defined using tuples, is there an equivalence between the two? Can we write a column vector as being equal to a tuple, perhaps in the column vector case this is just an alternative notation for a column vector and in fact not a tuple?
If there was an equivalence the difficulty is that both a row and column vector could in theory be written as a tuple, and as we know the matrices are not equal.
Best Answer
A vector is, definitionally, an element of a vector space; they may be tuples, but don't have to be. (Functions can form a vector space, for instance.)
Critically, a vector space requires additional structure: one needs to define an addition and scalar multiplication, and the scalars and tuple entries must come from a field. If we define the operations in the usual pointwise fashion, then, sure, all tuples of a particular fixed size are a vector space.
But as stated, tuples by themselves are not vectors; simply tuples.
The choice of a row vector versus a column vector is not particularly noteworthy and important, and the operation of transposition lets one easily slide from one perspective to the other. It's only really important for the sake of matrix operations and certain properties associated with matrices.
I would hesitate to say there's any sort of "equivalence" going on here, however, prior to some definition of said equivalence being given. Tuples are just natural examples of vectors, but not equivalent to them. Row vectors and column vectors are not the same thing, but concerns focused on one are often easily translated to the other.