Primarily written for a proof-based linear algebra course, this textbook lays down the necessary theoretical foundation of linear algebra. Our aim has been to cover the senior undergraduate mathematics course, meeting the curriculum requirements of most universities. To this end, this book has taken shape from our lecture notes, continuously modified and improved year after year through interaction with students in the classroom and during individual discussions. It is supported by the cumulative weight of our combined teaching experience, which ensures that the final product is uniquely equipped to cater to the needs of students as well as teachers. Another distinguishing feature of the book is that it offers a unified treatment, encompassing both the basics of computations with matrices and the abstract theory, and accentuating the overlaps between the matrix-oriented viewpoint and abstract linear algebraic concepts whenever possible. It is written in an accessible style, with careful thought to the motivation behind new ideas, and close attention to the exploration of difficult definitions and results, once they have been formally stated. In an effort to make our book more inviting and stimulating than others, we have incorporated interesting tidbits from the history of the concepts we have tackled into our discussions of those concepts. The book comprises eight chapters, opening with the definition and examples of Vector Spaces. It then traces a path through Bases and Dimensions, Linear Transformations, Matrix Representation of a Linear Transformation, Dual Spaces, Diagonalization and Inner Product Spaces. Contents 1. Vector Spaces 2. Basis and Dimension 3. Linear Transformations 4. Matrices of Linear Transformations 5. Dual Spaces 6. Diagonalization 7. Inner Product Spaces 8. More on Inner Product Spaces Appendix I: Echelon Form Appendix II: Applications of Diagonalization Appendix III: Least Squares Approximations Appendix IV: Minimal Solutions to System Appendix V: Rigid Motions and Conic Sections Selected Bibliography Index About the Authors Dr. Dinesh Khattar, erstwhile Principal (2015 to 2018), serves in the post of Professor at the Department of Mathematics, Kirori Mal College, University of Delhi. He was awarded the gold medal for topping in his B.Sc. As well as M.Sc. Exams at Delhi University. He was the recipient of the Dr. S. Radhakrishnan Memorial National Teacher's Award in 2015, for his contributions to the field of education. He was also awarded the prestigious Commonwealth Scholarship for pursuing research in the UK. He is actively involved in research and has presented papers at various international conferences across the globe. Dr. Khattar has been a member of the curriculum development committee for B.Sc. And M.Sc. Programs at a number of universities, including the University of Delhi. He has authored numerous books on mathematics, among which Group Theory as well as Ring Theory have been published by Ane Books. Dr. Abhishek Tandon is a senior faculty member at Department of Management Studies, Shaheed Sukhdev College of Business Studies, University of Delhi and Visiting Faculty at Department of Business Economics, University of Delhi. He completed his education from Kirori Mal College, University of Delhi and holds a PhD Degree in Applied Mathematics from University of Delhi. He has published extensively in reputed international journals. He was awarded as Best Teacher by Government of NCT of Delhi in 2016. He was also awarded prestigious UGC BSR meritorious students’ scholarship.. Dr. Neha Agrawal completed her education from Kirori Mal College, University of Delhi, and pursued her M.Phil. And Ph.D. from the Department of Mathematics, University of Delhi. Her areas of interest are Nonlinear Dynamical Systems and Chaos Theory. She has been working as an Assistant Professor at the Department of Mathematics, Kirori Mal College since 2012. She has published several research papers in esteemed international journals,