TITLE: Why are linear transformations important?
QUESTION [9 upvotes]: How important are linear transformations in linear algebra? In some texts linear transformations are introduced first and then the idea of a matrix. In other books linear transformations are relegated to being an application of matrices. What is the best way of introducing linear transformation on a linear algebra course? How do we motivate students to study transformations as part of linear algebra? What is their real impact?

REPLY [3 votes]: I would simply say 'geometry' can be a good motivation. Mention rotations, reflections, similarity trnasformations, projections to a subspace.. Roughly speaking they are the geometrical transformations that keep the origo and take lines to lines. I understood much better matrices when I could imagine some geometry behind..