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.
Given a vector space, V, over field F (typically R or C) we can define its "dual space", V\*, as the set of all linear functions from V to F. It can be shown that V and V\* are isomorphic so V\* is also a vector space.

In particlar, if V is finite dimensional, say dimension n, it is customary to represent each vector as a vertical n-column and each linear function as a horizontal n-row. For example, if V is three dimensional we can write a vector as $v= \begin{bmatrix}x \\ y \\ z \end{bmatrix}$ and a linear function as $f= \begin{bmatrix}a & b & c \end{bmatrix}$. Then the operation of function f on vector v can be written as a matrix product:
$f(v)= \begin{bmatrix}a & b & c \end{bmatrix}\begin{bmatrix}a & b & c \end{bmatrix}= ax+ by+ cz$.

This is, of course, just notation.