Have you done matrices and determinants?

[loonychune]

I did post earlier about creating a course on linear algebra for myself but got no reply, so i probably will work through the chapter of a book by Riley, Hobson and Bence.

However, if you could, please express how you satisfied yourself with such things as matrices and determinants as useful tools for solving linear equations and physics problems...

 

[h0dgey84bc]

I did post earlier about creating a course on linear algebra for myself but got no reply, so i probably will work through the chapter of a book by Riley, Hobson and Bence.

The Riley Hobson and Bence book is really good for linear algebra.

However, if you could, please express how you satisfied yourself with such things as matrices and determinants as useful tools for solving linear equations and physics problems...

Are you asking are matrices and determinants useful in physics?
Matrices and determinants are ubiquitous in physics, even in classical mechanics. Of course Quantum Mechanics is built on linear algebra, and since most of modern physics contains QM then most of modern physics has some form of linear algebra content. If you intend to study General relativity then you will need familiarity with matrices before meeting their generalisation, the tensor.
You don't really need all the formal proofs, just knowledge of how to apply the results, but never the less you'll be using them so often it's nice to see their origin.

 

[HallsofIvy]

Matrices and determinants should play only a small part in a Linear Algebra class. Linear Algebra is about vector spaces and linear transformations between vector space.

Given a specific basis, a linear transformation can be represented by a matrix, but it should be always remember that that is only a representation, not the linear transformation itself.

 

[clope023]

I'm taking linear algebra right now, the 1st 2 chapters were basic matrix/determinant algebra; but now we're going into abstract vectore spaces in 2 and 3 space.

 

[loonychune]

Well in my original post I was a bit less 'fluffy' in title and post...

Riley,Hobson, Bence begins by talking about vector spaces, basis vectors, the inner product, useful inequalities and i suppose those things really that make more formal and generalise those aspects met in simple vector algebra courses (dot product etc.).

My problem is that there are reams of text on things like 'change of basis and similarity transformations', quadratic and hermitian forms etc. but the book doesn't really talk about things like, for homogeneous equations we require that the determinant of the matrix set-up = 0 for a unique solution, and only talks about cramer's rule which is all good and well for inhomogeneous N x N forms but i was wondering how to go about solving M x N forms and their discussion of the method of SINGULAR VALUE DECOMPOSITION seems arguably over-the-top and more something for reference (Boas does talk about DET = 0 for homogeneous equations but without proof and also leaves a lot to be desired when it comes to vector spaces and the like).

Any suggestions would be MUCH APPRECIATED :)

 

[h0dgey84bc]

My problem is that there are reams of text on things like 'change of basis and similarity transformations', quadratic and hermitian forms etc. but the book doesn't really talk about things like, for homogeneous equations we require that the determinant of the matrix set-up = 0 for a unique solution, and only talks about cramer's rule which is all good and well for inhomogeneous N x N forms but i was wondering how to go about solving M x N forms and their discussion of the method of SINGULAR VALUE DECOMPOSITION seems arguably over-the-top and more something for reference (Boas does talk about DET = 0 for homogeneous equations but without proof and also leaves a lot to be desired when it comes to vector spaces and the like).

Stuff like Cramers rule is hardly ever used in physics (directly anyway, results that stem from it yes), seriously it's one of those things I learned in first year and never ever seen again.All you actually ever use, is the determinant being zero for unique solution, and the result about how to find eigenvalues from characteristic equations. The stuff about vector spaces, Linear operators representation as matrices, similarity transformations, diagonalisation by sim transform to basis of eigenvectors, etc...is much more useful, than all the stuff regarding simultaneous equations.

 

[atyy]

However, if you could, please express how you satisfied yourself with such things as matrices and determinants as useful tools for solving linear equations and physics problems...

The linear algebra for solving linear equations and physics problems is almost completely different. For physics, the main use is in quantum mechanics, where the axiomatic definition of a vector space, bases, inner product, eigenvectors and eigenvalues are the most important bits (this is probably why HallsofIvy said matrices and determinants are not so important). In solving linear equations, matrices, determinants and singular value decomposition are more important. To solve a linear equation, you need to invert a matrix, and for large matrices, it turns out to be very challenging numerically because large values become very small (or something like that), and here you should probably consult Numerical Recipes in C (old versions are available online, I think).