# Thread: What to study after linear algebra

1. ## What to study after linear algebra

Hey MHF,

I've been teaching myself linear algebra for giggles. I have Strang's Linear Alegra and its Applications and I've been watching his lectures on youtube. I have 3 years of calc behind me (all of Anton's 8th edition). I've also read the first half of Murray Spiegel's Applied Differential Equations. (I stopped after the solution methods by use of series and before systems of ODEs.) What should I study next? I was thinking of doing PDEs, though I'm kind of bored with DEs. I'd be super grateful for a few suggestions =D

Kgm

2. I don't know if you're into physics, but if you are, quantum mechanics is a fantastic follow-up to linear algebra. Prerequisites for quantum mechanics (in my opinion) are:

Freshman and Sophomore-level physics.
Junior-level classical mechanics (although I survived without having had that first - it is a little better if you do it first).
Calculus 1, 2, and 3.
Differential Equations.
Linear Algebra.

That'd be my recommendation. If you're not into applications so much, you might go with Advanced Calculus, Real Analysis, Complex Analysis (a particularly beautiful subject), or even Abstract Algebra.

3. If I remember correctly, then Strang's book is not very theoretical. If you want to start study more rigorous mathematics I would suggest abstract algebra or real analysis. You may want to look at a slightly more theoretical book on linear algebra first though. Linear Algebra tends to be a nice bridge from the computational to the theoretical.

Complex Analysis is very nice, but you will have to accept some results from Real Analysis as you go.

4. Originally Posted by Kgm
Hey MHF,

I've been teaching myself linear algebra for giggles. I have Strang's Linear Alegra and its Applications and I've been watching his lectures on youtube. I have 3 years of calc behind me (all of Anton's 8th edition). I've also read the first half of Murray Spiegel's Applied Differential Equations. (I stopped after the solution methods by use of series and before systems of ODEs.) What should I study next? I was thinking of doing PDEs, though I'm kind of bored with DEs. I'd be super grateful for a few suggestions =D

Kgm
Everything in Linear Algebra is finite-dimensional. If you wanted, you could crank everything up a knotch and look at the infinite-dimensional analogues of Hilbert spaces and Banach spaces. It is, admittedly, graduate level stuff, but very interesting all the same.

This stuff can be found in the book functional analysis' by Rudin.

5. Originally Posted by Swlabr
Everything in Linear Algebra is finite-dimensional. If you wanted, you could crank everything up a knotch and look at the infinite-dimensional analogues of Hilbert spaces and Banach spaces. It is, admittedly, graduate level stuff, but very interesting all the same.

This stuff can be found in the book functional analysis' by Rudin.
I think you mean that a basic course in Linear Algebra only looks at finite dimensional spaces. You can certainly study infinite-dimensional vector spaces. This would still be considered Linear Algebra.

Personally I think that Functional Analysis is too big a jump from Linear Algebra. You may want to study it after you've been exposed to a bit more theoretical math.

6. ## Re: What to study after linear algebra

Originally Posted by Swlabr
Everything in Linear Algebra is finite-dimensional. If you wanted, you could crank everything up a knotch and look at the infinite-dimensional analogues of Hilbert spaces and Banach spaces. It is, admittedly, graduate level stuff, but very interesting all the same.

This stuff can be found in the book `functional analysis' by Rudin.
Yutaka Yamamoto's book "From Vector Spaces to Function Spaces: Introduction to Functional Analysis with Applications" is also a really good book to check out for that.