TITLE: Why use a matrix?
QUESTION [6 upvotes]: Why would I want to use a matrix?  It's good for organizing a few numbers but I can't find too much use for them.  Could someone explain?

REPLY [3 votes]: We use matrices to study linear transformations between vector spaces. In particular, suppose we have a linear transformation $T: V \to W$ between two vector spaces $V$ and $W$. Suppose we have an ordered basis of $V$ and an ordered basis of $W$. Then we can represent vectors in $V$ and $W$ as column vectors. We can then represent $T$ by a matrix.