Why do vectors often have 1 dimenision as 1 In textbooks and school, why is it that vectors are often shown to be n x 1 or 1 x n matrices? I have rarely seen other dimensioned matrices representing matrices. 

\begin{bmatrix}
    1 &amp; 2 &amp; 3 &amp; 4 &amp; 5\\
    3 &amp; 4 &amp; 5 &amp; 6 &amp; 7\\ 
  \end{bmatrix} is usually a matrix but

\begin{bmatrix}
    1 \\
    3 \\
    4 \\
  \end{bmatrix}
is usually a vector. Why?
Simplicity of notation. Using a 2x3 matrix allows you to represent 6 quantities with the understanding that there is some sort of "grouping" within those 6 quantities (2 sets of 3 quantities that are in some way "different"). When we work with vectors we are not usually concerned with such an internal grouping. Note, however, that there is nothing stopping you from viewing a 1xn set of scalars as a matrix or as a vector. You must simply make clear the type of mathematical object you are working with. After all, you could represent a 2x2 matrix as a column of 4 quantities, right? but that wouldn't be a very convenient or simple notation.
Yes, usually column vectors ($1\times n$ matrices) are used for vectors. But vectors are also very commonly regarded as such matrices, e.g. the matrix product $A\cdot v$ is everywhere used. 

For another example, we can also build a matrix (of range $1$) by the matrix product $v^Tv$.

So, in many cases it is extremely useful to apply matrix operations for (column and/or row) vectors simply regarding them as matrices.