Linear algebra is the branch of mathematics concerning finite or countably infinite dimensional vector spaces, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.
Linear algebra is central to both pure and applied mathematics. For instance abstract algebra arises by relaxing the axioms leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.
MIT offers the course in linear algebra which offers a thorough introduction to the subject. We have recommended it on the blog earlier. It is taught by Gilbert Strang, a teacher I hold in high regards for his ability to deliver a subject in an approachable way. The course consists of 34 lectures each being about 45 minutes as far as I know, plus a lot of exercises and answers for you to get it under your finger nails. I have seen the course many years ago when I was reading up on the subject and can recommend the course as something which will give you a gentle but thorough introduction to the topic.
The only thing I would recommend is that rather than you watching the tidbits of videos for each lecture, you watch the whole lecture which has been taught. This gives a much more coherent understanding of the topic. Officially Multivariable calculus is a prerequisite, but it also mentions that you should be able to understand the course without it. However, if you like to study mutlivariable calculus, just follow the link to some good resources.
As with most other basic math topics Khan academy has a long series on linear algebra. This is more focused on how to a apply the different formulas and what the different elements does than actually deriving the formulas and proving that they are true. I have used this resource to gain a better understanding of some topics and not least refresh some topics that I had nearly forgotten. I wouldn’t use this as a stand alone source if you are looking into linear algebra, but I would definitely use it as a supplement for most other sources to pick up on things that are difficult to understand when you are first introduced to them.
I find this book to give a good introduction to the topic of linear algebra. It is not very rigorous, so if you are looking for a book which takes on a more proof based approach this one is not for you. However, if you are looking for an introduction to the topic, then this is a good book for breaking into it. I know that many people recommend this book together with the MIT lectures recommended above. Another comment I have read, suggests to take a cup of tea and read the book slowly in order to really understand the concept. I think that is a more general comment to mathematics then to this book in general.