Many difficult science problems can handled using the powerful, yet easy to use, mathematics of linear algebra. Unfortunately, because the subject (at least for those learning it) requires seemingly arcane and tedious computations involving large arrays of number known as matrices, the key concepts and the wide applicability of linear algebra are easily missed. Therefore, before we equip you with matrix skills, let us give some hints about what linear algebra is. The takeaway message is
Linear algebra is the study of vectors and linear transformations.
In broad terms, vectors are things you can add and linear transformations are very special functions of vectors that respect vector addition. To understand this a little better, lets try some examples. Please be prepared to change the way you think about some familiar mathematical objects and keep a pencil and piece of paper handy!