 This is your instructor for the course applied linear algebra. I am Dhoipayan Mukherjee. I am associated with the Department of Electrical Engineering, the group that is control and computing. The course that I'm about to deliver to you now is a core course for our postgraduate students that is our MTech students, as well as a very useful course for all our PhD students, particularly from the communications group, as well as from the control and computing group. So what do we do in this course? To summarize briefly, this is a second level course for undergraduate students, as well as we see a good number of undergrad students also crediting this course. Even if you are not very familiar with the basic tenets of linear algebra, but are somewhat familiar with matrices, that is still a good place to start on your journey in this course. So what do we do? This is a slightly more abstract take on matrices and vectors that you are also familiar with in the context of Euclidean spaces. So we look at more abstract objects such as rings and fields to start with. However, we never lose sight of the basic principles that you already are aware of, which are matrices and vectors in Euclidean spaces. However, all our proofs, all our ideas are related to more abstract objects such as more abstract vector spaces, some of which could even be infinite dimensionals. Speaking of which, we give you a much more formal definition and take on what these ideas of dimensions are, basis, what is a basis that is, how linear transformations can be represented as matrices through some coordinate representation. And we show you essentially some unknown objects in terms of known objects, that is the goal of this course. So you might ask, where is this application part coming in? We do not delve into specific applications as such, but by the time we are done with this course, you shall be in a very good position to apply the concepts that you learn during this course to whatever your domain is, that is our hope. We embark on a journey towards linear transformations. We see the celebrated rank nullity theorem and its proof. Essentially, most of the results that you will see will come with rigorous proofs. So if you are a researcher who reads research papers pretty often, but find it a little tiresome to wrap your head around the proofs, this is a course for you. If you are a postgraduate student, a master's level student, whether from engineering or from the basic sciences, you will still find something in this course that is going to be useful to you, because again, the rigor level is never going to be compromised. At the same time, we do not necessarily regurgitate material from any one textbook as such, but rather we mix and match stuff so that it is much more palatable and digestible. At least that has been the goal throughout this course during the last three, four years when I have been offering it to our students. So by the time we are done with this course, you will be familiar with not just how to get the Jordan canonical form of a matrix, but also why the Jordan canonical form must exist. You will be in a position to solve linear time invariant differential equations. You will also be in a position to apply the ideas of best possible approximations using the ideas of inner product spaces. You will also be in a position to learn about the ideas of dual spaces, double duals, quotient spaces. And I am sure if you have any involvement, if you have had any involvement with control theory at an advanced level, such as multivariable control systems, then this course will definitely pave the way towards a better understanding of such subjects. If you are from a communications background, from signal processing, even then you will find the contents of this course very useful. If you are from the industry, but you are actively involved in research, I am sure you will have to read papers pretty often. As I said at the beginning, even then this course will sort of help you shed the baggage of proofs and theorems and the formal language that papers come with. And you will be able to see them clearly, you will be able to get to the idea, you will be able to cut to the chase and get to the idea of the results without being worried about the language and the paraphernalia that comes with it. So all in all, I hope you all enjoy this course and you learn something out of it. And not just that this course will help you in thinking mathematically, but also because the premise of all mathematical thought is based on very strong logic. So I hope that as a person as well, it will help you in thinking in more logical terms than what you would have done had you not embarked on this journey with me. So I hope you will enjoy this course. Happy learning. Thank you.