This book is the first part of a three-part series titled Problems, Theory and Solutions in Linear Algebra. This first part contains over 100 solved problems and 100 exercises.

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This book is the first part of a three-part series titled Problems, Theory and Solutions in Linear Algebra. This first part contains over 100 solved problems and 100 exercises on vectors, matrices, linear systems, as well as linear transformations in Euclidean space. It is intended as a supplement to a textbook in Linear Algebra and the aim of the series it to provide the student with a well-structured and carefully selected set of solved problems as well as a thorough revision of the material taught in a course on this subject for undergraduate engineering and science students.

**Marianna Euler**

Marianna Euler is an associate professor of mathematics at Luleå University of Technology, where she is teaching several undergraduate mathematics course for Engineering students, including courses in linear algebra, differential equations and linear analysis. She is an active researcher in the subject of nonlinear partial differential equations and Lie symmetry transformation groups on which she has published over 40 research articles.

**Norbert Euler**

Norbert Euler is a full professor of Mathematics at Luleå University of Technology in Sweden. He is teaching a wide variety of Mathematics courses at both the undergraduate and graduate level and has done so at several universities worldwide for the more than 25 years. He is an active researcher and has to date published more than 70 peer reviewed research articles in Mathematics and Mathematical Physics journals and is the co-author of several books. In his research he specializes on the subject of Nonlinear Differential Equations (both ordinary- and partial differential equations) in Mathematical Physics, of which he is studying the equations' integrability properties and methods of exact solutions by algebraic and geometrical means using, e.g. the Lie symmetry Algebra and Lie Transformation Group structures. He is also involved in editorial work for some international journals and he is the Editor-in-Chief of the Journal of Nonlinear Mathematical Physics. For more information, please visit his personal website at http://staff.www.ltu.se/~norbert/

- Vectors, lines and planes in R3
- Vector operations and the dot product
- The cross product
- Planes and their equations
- Lines and their parametrizations
- More on planes and lines
- Exercises

- Matrix algebra and Gauss elimination
- Matrix operations of addition and multiplication
- The determinant of square matrices
- The inverse of squarematrices
- Gauss elimination for systems of linear equations
- Square systems of linear equations
- Systems of linear equations in R3
- Intersection of lines in R3
- Exercises

- Spanning sets and linearly independent sets
- Linear combinations of vectors
- Spanning sets of vectors
- Linearly dependent and independent sets of vectors
- Exercises

- Linear transformations in Euclidean spaces
- Linear transformations: domain and range
- Standard matrices and composite transformations
- Invertible linear transformations
- Exercises