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?

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?

I have read Linear Algebra from Axler's "Linear Algebra Done Right". So I 'd say that Linear Algebra is all about linear transformations. Matrices are secondary. In Axler's book it is briefly discussed how many solutions a system of equations has and this problem can be solved using linear transformation. I don't know what would be the best way to introduce LT to beginners. If you follow Axler's book I think all your questions are answered simultaneously.

Here's an application of rotation matrices, which is one of the more important kinds of linear transformations. Rotation matrices are used in Lie Algebras, which show up in the solution of differential equations.