What is the benefit of writing lecture notes for an introductory course vs using a textbook?
As someone who's tried to do both, there are some very valid reasons to prepare "formal" lecture notes.
The primary reason why you'd want to create your own notes is that for many courses, a single good text is not available, and as a result, the instructor has to cobble together material from a number of different sources to produce a coherent set of lecture notes—or recommend that students work with multiple source texts. (Given the out-of-control nature of textbook prices, the latter alternative is unlikely to work out well.)
If you have a single-text class, it may not be necessary to provide students an additional set of notes, provided your lectures stick to the main text material. However, if you bring in alternative or additional topics into your lectures, you may want to include notes for those topics, and refer students to the textbook for places where you follow the "standard" outline.
As you are teaching linear algebra, I will use it as an example. Gilbert Strang teaches/taught linear algebra at MIT. He also wrote the textbook Introduction to Linear Algebra. You might expect the course to follow along perfectly with the textbook. If you look at the syllabus from when Strang was teaching the course or now what you will see is there are significant deviations. It goes section: 3.6, 8.2, 4.1-4.4, 8.5, 5.2-5.3, 6.1-6.2, 6.6, 8.3, 6.3.
If the author of the textbook cannot even happily follow his own ordering of the topic, it is not surprising that many teachers feel the need to create their own notes that go in the order and cover the material in the depth that they want.
I have written lecture notes for a couple of courses, and in one of those cases I assigned a textbook as well. Here are some of my reasons.
For a course in which I have some flexibility as to the content, I may find that no single textbook includes all the material I want to teach. Asking students to buy three or four books is rather obnoxious, especially if for some of them I will only be referring to a few pages. When I write lecture notes, I can include exactly the material that I want to include in the course.
Even when I am generally following a textbook, I often find places where I want to go off on a tangent, or discuss additional related material, or maybe just cover the same material with a different treatment. I feel like me teaching a class should add value beyond the student reading the textbook, and one way to do that is to tell students how I personally think about and understand the material in question. It can be helpful for the student to have that in writing. In part this is probably arrogance, but I really do feel I have insights to offer that are not contained, or not as well expressed, in even the most "well-written" textbook.
If I am going to need fairly detailed and precise notes to lecture, I might as well type them - they'll be neater, easier to read, and I can refer to them next time I teach the course. If I'm going to that trouble, I might as well make them available to the students.
I find that writing lecture notes for an audience other than myself is a really effective way to teach myself material, and understand it at a deeper level. It very often leads me to new insights on something that I thought I understood.
Written lecture notes that are posted on a website can be helpful to anyone in the world, not just the students in my course. I've been able to answer questions on MathOverflow and Math.SE by pointing people at my lecture notes.
For high-level courses (especially graduate topics courses), there may not be any textbook on the relevant material - I am assembling it from the research literature. But in order to use material from a research paper in a course, I usually have to rewrite a lot of it - filling in background and omitted details, and so on. So it becomes lecture notes.