If vectors are a special case of a matrix, then how is A transpose B ≡ A ∙ B (dot product)? If I have two vectors of equal dimensions like 3x1 lets say, how am I able to multiply it with another vector of 3x1 dimension? Its not possible in matrix operation.
Edit: Seems like most of you didn't get my question. If vectors ARE matrices and A and B has the same dimensions (3x1), it should not have been possible to multiply no?
 A: The dot product of two vectors is defined as
$$\bar{v} . \bar{u} = v^Tu$$
This would result in a $1 \times 1$ real number
A: Multiplication $AB$ between an $m \times n$ matrix $A$ and a $k\times l$ matrix $B$ is only defined if $n=k$ and then gives a $m\times l$ matrix. Vectors are $n\times 1$ matrices, with $n=3$ in your case, and matrix multiplication between two vectors is therefore not defined (unless $n=1$), but multiplication of a vector and its transpose is as you have $(1\times n) \times (n\times 1)$ giving a $1\times 1$ matrix that is a number.
A: I'm going to explain this using a classical view of vectors, so without vector spaces.
Vectors are not special cases of matrices! It's just our notation for both that's similar. But the matrix $\begin{pmatrix}1\\2\\3\end{pmatrix}$ and the vector $\begin{pmatrix}1\\2\\3\end{pmatrix}$ have different meaning.
The vector stands for a translation of space: You can apply it to a point to shift the point in a certain direction, and the entries tell you to shift the point 1 unit to the right, 2 units forward and 3 units upward (assuming our coordinate system is aligned in a fitting way).
The matrix, however, is something you apply to a vector. A boring one, mind you: a vector with just one entry. It sends the one component vector to a three component vector according to the usual rules of how to apply matrices to vectors.
Now, the dot product tells us something about how closely related two shifts described by a vector are: the larger the dot product, the more closely aligned their directions are. The matrix product, on the other hand, tells you how to combine their corresponding transformations: if the matrix $A$ encodes one transformation, while $B$ encodes another, then the matrix $BA$ encodes the transformation we'd obtain by first applying $A$, then $B$.
In essence, vectors shift points, matrices transform vectors. The dot product compares shifts, while the matrix product "chains" transformations. From this point of view, it's unsurprising that they don't behave the same way, since they do completely different things. On the contrary, it might be surprising to see that the dot product does behave kind of like matrix multiplication. The reasons are quite deep, and have to do with the fact that the dot product can also be interpreted as a transformation of a vector: if we fix a vector $u$, then we can transform another vector $v$ to get a number by taking their dot product. Such transformations can be expressed using matrices, and the corresponding matrix is $u^T$.
A: If $A$ and $B$ are two matrices of $3\times 1$ you can't multiply them. The product of matrices require that the number of columns of the first factor to be the same that the number of rows of the second factor:
If $C \in M_{k\times \color{green} m}, D \in M_{\color{green} m \times n}$ then $CD \in M_{k\times n}$
In the case of two vectors, $3\times 1$ the dimensions doesn't match.
$A \in M_{3\times \color{red} 1}, B \in M_{\color{red} 3 \times 1}$
