Difference in Notation for Vectors in Linear Algebra & Multivariable Calculus Often in Linear Algebra we see vectors depicted either in Column or Row Form as :
Linear Algebra : Vector in Row Form
$$ \vec{V}^{\,} = \left[x_1,\ldots,x_n\right]$$
OR 
Linear Algebra : Vector in Column Form $$ \vec{V}^{\,} = \left[\begin{array}{c}x_1\\\vdots\\x_n\end{array}\right]$$
But in Multivariable Calculus, often, angle brackets/chevrons are used to represent vectors like so :
Representation of a Vector in Multivariable Calculus:
$$ \vec{V}^{\,} = \left<x_1,\ldots,x_n\right>$$
Is there any fundamental difference between these three representations of Vectors or is it purely a difference in notation and convention?
 A: Whether you use a bracket or a chevron depends on the convention of the book you may be reading.
However, it is important to consider if we are dealing with a column or a row vector. As far as I'm concerned, if we plainly say vector, that means that it is written in the form of a column. You should be careful when dealing with this, as some operations with the "same" vector but written in different ways may lead to different results.
For example:
$$\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]
\left[\begin{array}{ccc} 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$
However, if we write the first vector as a row:
$$\left[\begin{array}{ccc} 1 & 0 & 0 \end{array}\right]\left[\begin{array}{ccc} 0 & 1 & 0 \end{array}\right]=0$$
Also, some operations would not make sense if we expressed vectors in one way or another. For example, the product
$$\left[\begin{array}{ccc} 2 & 1 & 0 \\ 8 & 0 & 1 \\ 3 & 1 & 1 \end{array}\right]\left[\begin{array}{c} 1 \\ 1 \\ 2 \end{array}\right]$$
is well defined and can be executed with no further complications. However, the product
$$\left[\begin{array}{ccc} 2 & 1 & 0 \\ 8 & 0 & 1 \\ 3 & 1 & 1 \end{array}\right]\left[\begin{array}{ccc} 1 & 1 & 2 \end{array}\right]$$
makes no sense and can't be done.
