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: 
You are confusing notation and coordinate representation of vectors. That is, a vector space is an algebraic structure defined using four objects $(V, \mathbb{F}, \oplus ,\odot )$ where $(V, \oplus)$ is an Abelian group, $(\mathbb{F},+,\cdot )$ is a field and

$$
\odot :\mathbb{F}\times V\to V,\quad (\lambda ,\mathbf{v})\mapsto \lambda \odot \mathbf{v}
$$

is a function named scalar multiplication, such that the following conditions holds:

$$
\lambda \odot (\mathbf{v}\oplus \mathbf{w})=(\lambda \odot \mathbf{v})\oplus (\lambda \odot \mathbf{w}),\quad 1\odot \mathbf{v}=\mathbf{v},\quad 0\odot \mathbf{v}=\mathbf 0
\\(\lambda +\mu)\odot \mathbf{v}=(\lambda \odot \mathbf{v})\oplus (\mu \odot \mathbf{v}),\quad (\lambda \cdot \mu )\odot \mathbf{v}=\lambda \odot (\mu \odot \mathbf{v})
$$

If the vector space have finite dimension (say dimension $n$), then there exists some list of linear independent vectors $\mathbf{v}\_1,\ldots ,\mathbf{v}\_n$ such that for every $\mathbf{w}\in V$ there exists scalars $\lambda \_j\in \mathbb{F}$ such that

$$
\mathbf{w}=(\lambda \_1\odot \mathbf{v}\_1)\oplus \ldots \oplus (\lambda \_n\odot \mathbf{v}\_n)
$$

Then, using the previous list as a basis of $V$ we can represent the vector $\mathbf{w}$ as $(\lambda \_1,\ldots ,\lambda \_n)$, that is, there is a bijective map $\phi :V\to \mathbb{F}^n$ such that to each vector in $V$ gives a coordinate representation $(\lambda \_1,\ldots ,\lambda \_n)$, what is an element of $\mathbb{F}^n$.

So, for any Euclidean space $V$ of dimension $n$ we directly use the coordinate representation given by elements of $\mathbb{R}^n$. Now, as a notation we can represent any element of $\mathbb{R}^n$ by the standard notation for $n$-tuples, that is $(\lambda \_1,\ldots ,\lambda \_n)$, or using a matrix-like vertical notation

$$\begin{pmatrix}
\lambda \_1\\ \vdots \\ \lambda \_n
\end{pmatrix}$$

However both notations represent the same vector $\mathbf{w}\in V$, but we choose some or other notation depending on the context to make things easier, by example if $M$ is an $n\times n$ matrix then we choose the vertical notation to represent the action of $M$ by the left to some vector, in this case

$$
\begin{pmatrix}
M\_{1,1}&&\cdots &&M\_{1,n}\\\vdots && &&\vdots \\M\_{n,1}&&\cdots && M\_{n,n}
\end{pmatrix}\begin{pmatrix}
\lambda \_1\\ \vdots \\ \lambda \_n
\end{pmatrix}
$$

However, to write the coordinates of $\mathbf{w}$ inside a text is preferable to use the notation $(\lambda \_1,\ldots ,\lambda \_n)$ instead. I hope you see it more clear now.
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.