# 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?

• If you study arbitrary transformations you are not doing linear algebra anymore. Sep 25, 2012 at 10:07
• The theory of linear transformations is really important to understanding quantum mechanics, even at the undergraduate level. That's why I find it odd that undergraduate linear algebra is mostly 3x3 matrices, determinants, solving coupled algebraic equations (like we did in high school), and vectors in $R^3$. With that said, I didn't appreciate the concept of vectror spaces when I took linear algebra, so perhaps understanding real linear algebra takes some mathematical maturity. If you're not aware, take a look at "Linear Algebra Done Right." Jul 8 at 11:33

• 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: $$\forall v \in V, \forall \lambda \in K, f(\lambda v)=\lambda f(v).$$