# Why are linear transformations important?

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?

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If you study arbitrary transformations you are not doing linear algebra anymore. –  Raskolnikov Sep 25 '12 at 10:07

May be it does not exist in english textbooks Haha ! I'm sorry, I translated from french... Here's the exact definition of what I call linear application in my answer: Let $V$ and $W$ be two finite dimensional vector spaces over a field $K$, a linear application $f$ is a groups morphism from $(V,+)$ to $(W,+)$ that has the following property: $$\forall v \in V, \forall \lambda \in K, f(\lambda v)=\lambda f(v).$$ –  mak Sep 25 '12 at 10:51