Question: When dealing with Euclidean vector spaces (such as those in physics) we consider that a vector $A$ with a tuple of scalar components $(a,b,c)$ is equal to a vector $B$ with the same components, however, we make a distinction between column and row vectors (namely that one is the transpose of the other) is it that for a euclidean vector we just decide on one way of 'representation' and the distinction is more for dealing with matrices? Because this suggests that given a row vector $A$ and Column vector $B$ with the same components they are not equal. Is it a form of notation for vectors or are we essentially writing the components in a matrix?

If we define a vector $a=(a\_1,a\_2)$ as being represented as a column and row vector, then for $a$, $Ma≠Ma$ as one is defined and the other is not.

Answer: 
Let $n$ be a positive integer.

* A row vector with $n$ real components is a matrix with one row and $n$ columns, i.e. an element of $\mathbb R^{1\times n}$.
* A column vector with $n$ real components is a matrix with one column and $n$ rows, i.e. an element of $\mathbb R^{n\times 1}$.
* An element of $\mathbb R^n$ is a list of $n$ real numbers.

Of course we identify lists, row and column vectors through "obvious" bijections. For example, given a list$$(x\_1,\ldots,x\_n)\in\mathbb R^n$$ we can set $$\forall i:x\_{i,1}:=x\_i$$and $$\begin{pmatrix}x\_{1,1}\\\vdots\\x\_{n,1}\end{pmatrix}\in\mathbb R^{n\times 1}$$is the column vector associated to the list.
The answers above already well clarify that there is an isomorphism
between lists (n-uples), row vectors and column vectorsand thus the choice is a matter of convenience.

In linear algebra the column notation is more common because more conforming to the way we write
systems of linear equations.  

Instead the row notation is common practice in Markov chains just because the matrix then looks tha same as
the usual representation of a transition matrix as row $\to$ column.

But interestingly, since you are studying physics, later on you will face with the important concept of
[co-variant and contra-variant](https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors) vectors,
dealing with which is useful to use both conventions together, while keeping them distinct.