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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Linear transformations, if you mean linear applications, are fundamental in linear algebra. Actually, pretty much all the theorems in linear algebra can be formulated in terms of linear applications properties. Moreover, linear applications are morphisms which preserve the vector space structure and linear algebra is the study of vector spaces and for a big part the study of their endomorphisms. Endomorphisms are applications which are linear and associate vectors from one vector space to vectors in the same vector space. In general, every (good) algebra course talking about a certain structure (it could be groups, rings, fields, modules, linear representations, categories...) always start by defining the structure and its axioms, then defining sub-structures, and then morphisms that preserve that structure. In finite dimension, vector spaces are convenient because their scalars are elements of a field and they [the vector spaces] have a base, i.e. a family of vectors that are linearly free and generate any other vector. This property allows to represent endomorphisms as a table that gives you how you transform the vectors of that base into vectors of another base (this is theorem actually). Having this information is enough because you can reconstruct any other vector's image by linear combination and the properties of linearity of the endomorphism. Matrices thus definitely come after linear transformations as they are only a representation of them up to the choice of a base for the vector spaces. For linear applications that are from one vector space to another of different dimension (if it's the same dimension, the two vector spaces are isomorphic and you have an endomorphism), the matrix is rectangular because the two bases don't have the same cardinality (i.e. not the same dimension).
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..