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.matqkks said: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?
A linear transformation is a mathematical function that maps a vector from one vector space to another vector space, while preserving the structure of the vector space. It is a fundamental concept in linear algebra that is used to study and solve various real-world problems.
Linear transformations have a significant impact on linear algebra as they provide a way to transform and manipulate vectors and matrices in a systematic and structured manner. They help in solving systems of linear equations, finding eigenvalues and eigenvectors, and understanding the geometrical properties of vector spaces.
The primary motivation behind studying linear transformation in linear algebra is its wide range of applications in various fields, such as engineering, physics, computer science, and economics. It also provides a solid foundation for understanding more advanced concepts in linear algebra and other branches of mathematics.
Some common examples of linear transformations include rotations, reflections, shearing, scaling, and projection. In linear algebra, these transformations are represented by matrices and can be applied to vectors and matrices to produce new vectors and matrices.
A transformation is considered linear if it satisfies two properties: preservation of addition and preservation of scalar multiplication. This means that the transformation should preserve the operations of addition and scalar multiplication between vectors. In other words, the transformation should follow the rules of vector addition and scalar multiplication.