\section{Coordinates}
Vectors in $\bR^n$ can be thought of as \emph{column} matrices,
while elements of $\left(\bR^n\right)^*$, called \Def{covectors}
can be thought of as \emph{row} matrices.

The main reason for this is that way the evaluation pairing of a
covector with a vector is simply matrix multiplication.

More deeply, this reflect the isometry $\Hom(V,W) = V^* \otimes W$ for
finite dimensional vector spaces, and illuminates the structure of
linear transforms.

You can think of a linear transform as a matrix, which you can break
up as a bunch of column vectors, or a bunch of row vectors.

The column vector POV says:\\
``a linear transform is determined by the
\emph{vectors} in $W$ that it sends a basis of $V$ to''

The row vector POV says:\\
``a linear transform is determined by the \emph{coordinates}
  in $W$ that it \emph{evaluates} vectors in $V$ to''

Also, note that given a row and a column, if you multiply them the
opposite way, you get a matrix! (And not just any matrix -- a rank~$1$
one.) This story goes on for a while; we won't pursue it here.
