# How Advanced is 'Elementary' Linear Algebra?

I'm planning to start a bachelor's program next year, and it looks like I'll be required to take courses in Linear Algebra as prerequisites for the classes I'm most looking forward to.

Several years ago I took calc courses, but it was extremely difficult for me, and I have not done any complex math in nearly a decade. I'd be fortunate to remember how to do a quadratic equation.

Quick research tells me that 'Linear Algebra' is more complicated than I would think at a glance. Do I actually need to revisit calculus? What other specific subjects will prepare me to handle university-level math again? I'm willing to get text books and tutors, as I have nearly a year to prepare.

Thank you!

• What are the classes you're looking most forward to? You don't need calculus for linear algebra, but you will need calculus for virtually everything else. "Elementary" linear algebra is certainly more involved than "doing a quadratic equation" but still probably one of the easier, if not the easiest, course in undergraduate maths. – Damian Reding Sep 11 '15 at 16:02
• The absolute beginning college Math courses are Calc I, Calc II, Calc III and Differential Equations. Some programs will combine Linear Algebra with differential equations or even Calc III. However, most of us take Linear Algebra immediately after the above four. – John Douma Sep 11 '15 at 16:09
• @JohnDouma That sounds CRAZY. You waited until your FIFTH semester to take linear algebra? I took it in my first semester at the same time as Calc I. How do you even do Calc III or Diff EQ without any knowledge of vectors/ matrices/ etc? – user269351 Sep 11 '15 at 16:10
• @user269351Were you an advanced student or was your Linear Algebra course watered down. My Linear Algebra course started with the definition of a vector space. I really doubt someone who struggled with calculus has the mathematical maturity to understand that. – John Douma Sep 11 '15 at 16:14
• Linear Algebra is a prereq for Diff EQ and Calc III at my school so I could have taken it my second semester, but I just got it out of the way my first semester. We didn't cover as much as I would have liked (but then there is an upper division class for that), but we also included the definition of vector spaces. I don't see how memorizing a few axioms is hard. Then again, I didn't struggle with calculus either, so IDK how non-mathematically inclined students fared. – user269351 Sep 11 '15 at 16:16

Calculus would appear in linear algebra if at all only in the form of examples. For example, differentiation is linear and is not one-to-one.

Some adeptness in basic algebra is essential.

Some ability to do mathematical reasoning will also matter. How does one acquire that? Here unfortunately I'm not sure which books to recommend. This is mostly stuff I learned in 8th through 11th grades, and partly in various undergraduate math courses that were not primarily concerned with how mathematical reasoning is done.

I personally didn't find my first course in linear algebra very hard at all. It was very algorithmic so I didn't have to think too hard or be too creative to solve any of the problems. But that doesn't mean that you will have the same experience. I know there were other people in my class who thought it was difficult.

As for prereqs, you probably won't need calculus. At the very most I could see your professor including Wronskians which will require you to be able to differentiate a function. But it'll probably be functions which differentiate easily, like polynomials and trig functions. What you will need is a pretty solid understanding of high school algebra and a little bit of geometry -- mostly analytic geometry (using coordinates) rather than synthetic geometry (using just the axioms).

When I learnt about vectors in high school over twenty years ago, the main two operations used were the inner product, and the cross product.

It's easy to see (in hindsight!) that the cross product can't be quite right...this is because it's only defined in 3-dimensions; which unlike the inner product is defined for all spaces; and in fact, in more advanced treatments it's replaced by the wedge product.

The other distinction that's nots usefully emphasised is duality; that is the notion of a form; forms are dual to vectors, and admit all the same operations; and in fact, taking the dual of a form just gives you back the vectors.

This is useful since forms appears later in calculus on manifolds: ie differential geometry; and in physics - for example, using the the technology of forms the four maxwell equations are reduced to two; and also in that form we can ask them to hold not just in out 3d Euclidean space, but on any manifold of arbitrary shape and dimension.

But more, the stokes theorem replaces and generalises various theorems of the same form - Gauss and Green; and not only that it looks and is simpler.

All this suggests that getting to grips with forms as early as possible is useful; and in fact there is a free e-textbook which uses this to teach linear algebra: I forget the name, unfortunately ...

Taking a course in multivariate calculus helped a lot with some concepts in my opinion, but it wasn't required. I haven't used linear algebra in any subsequent courses yet, unless you count some of the theory we studied doing proofs (which had nothing to do with LA).

We used Kolman/Hill Elementary Linear Algebra as a text (ISBN: 978-0132296540). It builds up from the basics pretty well and is not intimidating for a self study if you're interested.