Exploring the Differences: Linear Algebra vs Algebra in Mathematics and Physics

[zalba]

This is something that I haven't really found much info on. I'm a student attending McGill university, and I have a choice between taking Honours Applied Linear Algebra and Honours Algebra.

Linear algebra:
Mathematics & Statistics (Sci) : Matrix algebra, determinants, systems of linear equations. Abstract vector spaces, inner product spaces, Fourier series. Linear transformations and their matrix representations. Eigenvalues and eigenvectors, diagonalizable and defective matrices, positive definite and semidefinite matrices. Quadratic and Hermitian forms, generalized eigenvalue problems, simultaneous reduction of quadratic forms. Applications.

Algebra:
Mathematics & Statistics (Sci) : Linear equations over a field. Introduction to vector spaces. Linear maps and their matrix representation. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Real and complex inner product spaces. Diagonalization of self-adjoint operators.

Now, generally the physicists take the linear algebra while the mathematicians take algebra. I'm interested in theoretical physics (currently GR, string theory, and astro), but since I'm only second year that may change.

 

[SteamKing]

Eventually, as a theoretical physicist, you will need to take both courses. There is some overlap, according to the course descriptions. IMO, I would take linear algebra first.

 

[mathman]

This is something that I haven't really found much info on. I'm a student attending McGill university, and I have a choice between taking Honours Applied Linear Algebra and Honours Algebra.

Linear algebra:
Mathematics & Statistics (Sci) : Matrix algebra, determinants, systems of linear equations. Abstract vector spaces, inner product spaces, Fourier series. Linear transformations and their matrix representations. Eigenvalues and eigenvectors, diagonalizable and defective matrices, positive definite and semidefinite matrices. Quadratic and Hermitian forms, generalized eigenvalue problems, simultaneous reduction of quadratic forms. Applications.

Algebra:
Mathematics & Statistics (Sci) : Linear equations over a field. Introduction to vector spaces. Linear maps and their matrix representation. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Real and complex inner product spaces. Diagonalization of self-adjoint operators.

Now, generally the physicists take the linear algebra while the mathematicians take algebra. I'm interested in theoretical physics (currently GR, string theory, and astro), but since I'm only second year that may change.

Based on the descriptions, the linear algebra course seems to have everything in the algebra plus additional material (abstract vector spaces, inner product, Fourier series). If the algebra course is for mathematicians I presume there is a subsequent course containing that material.

 

[WannabeNewton]

Both the courses you mentioned seem quite applied. What about honors linear algebra for pure mathematicians, have you considered that by any chance? I'm sure McGill of all places would have such a class.

 

[lavinia]

i would take the first course for the Fourier series.

My personal opinion is that a separate course in Linear Algebra is a waste of time. I would learn it as you go along.

Also, nowadays theoretical physics is no less mathematically sophisticated as theoretical mathematics. If you want to be a string theorist you will need theoretical math and need to understand it, not just use it.

 

[Fredrik]

Linear algebra:

Algebra:

The topics you listed are all topics in linear algebra, so the one called "algebra" is really a linear algebra course. I really can't tell how they're different based on these descriptions. Is the difference that one of them is more proof oriented?

My personal opinion is that a separate course in Linear Algebra is a waste of time. I would learn it as you go along.

I strongly disagree with this.

 

[lavinia]

I strongly disagree with this.

Why not explain why? I think it would be useful.

 

[Fredrik]

Why not explain why? I think it would be useful.

It's mainly that linear algebra is so useful in quantum mechanics, and also in relativity. The matrix version of Lorentz transformations is much more elegant (and in my opinion easier to understand) than the component version. You need to understand inner product spaces, orthonormal bases and eigenvectors and eigenvalues, for everything in QM. When you study spin-1/2 systems, you need to understand the relationship between linear operators and matrices*. If you know this, it will be much easier to understand tensors in general relativity.

*) Physics students always struggle with this, because they are terrible at it, even though they have taken a linear algebra course. So I think it would be better idea to take the course twice than to not take it at all.

Also, the books on QM don't explain this stuff very well, if at all.

 

[zalba]

I believe that the Algebra course is more of a proof-based course, due to the fact that in the regular physics degree (instead of the math-physics one I am in), they have to take linear algebra. This is the continuation of the course for algebra that I have a choice of taking (though I think I'm leaning more towards the analysis course anyway):

Algebra 3: Introduction to monoids, groups, permutation groups; the isomorphism theorems for groups; the theorems of Cayley, Lagrange and Sylow; structure of groups of low order. Introduction to ring theory; integral domains, fields, quotient field of an integral domain; polynomial rings; unique factorization domains.