I've read this post and posted my own question, but I think I will write a more direct question related to this topic, I understand how row and column vectors can be used to represent vectors as answered in the question, but is it actually a vector, or a way of us trying to give the idea of components by putting them in a matrix? For example, Euclidian vectors have no concept of 'transpose', if the components are equal, its the same vector, yet row and column vectors are transpose of each other?
A question I have is say we have:
$y = \begin{bmatrix}
x_{1} \\
x_{2} \\
x_{3}
\end{bmatrix}$
Can we have also $y = [x_{1},x_{2},x_{3}]$?
as then we have
\begin{bmatrix}
x_{1} \\
x_{2} \\
x_{3}
\end{bmatrix} = $[x_{1},x_{2},x_{3}]$
and this gives $y=y^T$ which would be an incorrect result.
Best Answer
A vector is an element of a vector space. A vector space is a set that satisfies a few specific requirements.
The set of $m \times n$ matrices with entries from a field $\mathbb{F}$ is always a vector space over $\mathbb{F}$, so all $m \times n$ matrices are vectors, including $m \times 1$ and $1 \times n$ row and column vectors.
As it so happens, there are also relationships between vector spaces - for example, the transpose of an $m \times n$ matrix is an $n \times m$ matrix, so we can say that $^T: \mathbb{F}^{m \times n} \rightarrow \mathbb{F}^{n \times m}$ is a relation between those two vector spaces. There is also a relation between $m \times n$ matrices and a plain old vector from $\mathbb{F}^{mn}$ where you write the elements of the matrix out into the components of the vector one at a time. If the relationship:
Is bijective (i.e. it maps every vector from the domain to exactly one vector from the codomain and vice versa), and
Preserves all of the vector space properties on each side (e.g. $(v_1 + v_2)^T = v_1^T + v_2^T$),
then the relation is called an isomorphism, and the two vector spaces are said to be isomorphic, and when two things are isomorphic you can essentially treat them as the same thing - as long as you only care about properties that pass through the isomorphism.
So yes - column vectors and row vectors are vectors, and they are equivalent to normal Euclidean vectors, as long as you're just talking about them as vectors. They also happen to be matrices, so you can do matrix things with them, but you need more sophisticated language if you want to relate those things to each other consistently since "doing matrix things" isn't necessarily covered by the vector space axioms.
As a small side note, you may not have even realised but you've already been bamboozled by isomorphic vector spaces, because that's what happens when you connect things written as $(x, y)$ with arrows drawn on a blackboard. Technically, one of those is just a pair of numbers inside a set of brackets while the other is a geometric construction, but we tend to just take it as granted that they're the "same" vectors because all the operations we want to do with them behave nicely across the divide so there's no point in making a real distinction.