I agree that that appears to be bullshit. When I think back to my courses, I had one in ‘vectors and matrices’ which is roughly equivalent to early courses in what American universities call ‘Linear Algebra’. I think it included:
- a refresher on matrix multiplication and eigenvectors/eigenvalues for those who hadn’t learned them in high school
- inverting arbitrary matricies (done with the horrific formalism and rigour that first year undergraduates sometimes like to see)
- basis transformations
- special matrices (orthogonal, unitary, hermitian, etc)
- suffix notation (by far the best bit)
- bilinear/quadratic/sesquilinear forms
- tensor products/contractions/basis changes (but without using tensor notation—it was all done with suffix notation thankfully)
And an exam question might be like:
- invert or diagonalise a 3x3 matrix of nice numbers. Or do Gaussian elimination to a 4x4. (These would be warm-up/small questions). If you get eigenvalues, maybe make a geometric interpretation.
- do some algebra with some variables known to be some kind of special matrix to prove something. (I.e. apply a property of the special kind of matrix to do the algebra)
- do some suffix notation algebra (in particular using some kronecker deltas or antisymmetric tensors and associated identities)
- something about bilinear/quadratic/sesquilinear forms/algebra
They mostly tried to avoid bullshit exam questions that were just computation. There were some other opportunities to do a bit of matrix computation like solving/linearising/sketching certain ODE systems, or computing Jacobians.
In second year we had a course that was actually called ‘Linear Algebra’ which touched on (numeric) vectors and matrices but mostly talked about linear maps, had lines of juxtaposed symbols with juxtapositions having different meanings between different symbols, and mostly involved statements like ‘let V be a vector space over F and let e_1,e_2,…,e_n be a basis for V’. I think the exams did not involve any computation and maybe involved producing some of the proofs from the course or applying them to something.
- a refresher on matrix multiplication and eigenvectors/eigenvalues for those who hadn’t learned them in high school
- inverting arbitrary matricies (done with the horrific formalism and rigour that first year undergraduates sometimes like to see)
- basis transformations
- special matrices (orthogonal, unitary, hermitian, etc)
- suffix notation (by far the best bit)
- bilinear/quadratic/sesquilinear forms
- tensor products/contractions/basis changes (but without using tensor notation—it was all done with suffix notation thankfully)
And an exam question might be like:
- invert or diagonalise a 3x3 matrix of nice numbers. Or do Gaussian elimination to a 4x4. (These would be warm-up/small questions). If you get eigenvalues, maybe make a geometric interpretation.
- do some algebra with some variables known to be some kind of special matrix to prove something. (I.e. apply a property of the special kind of matrix to do the algebra)
- do some suffix notation algebra (in particular using some kronecker deltas or antisymmetric tensors and associated identities)
- something about bilinear/quadratic/sesquilinear forms/algebra
They mostly tried to avoid bullshit exam questions that were just computation. There were some other opportunities to do a bit of matrix computation like solving/linearising/sketching certain ODE systems, or computing Jacobians.
In second year we had a course that was actually called ‘Linear Algebra’ which touched on (numeric) vectors and matrices but mostly talked about linear maps, had lines of juxtaposed symbols with juxtapositions having different meanings between different symbols, and mostly involved statements like ‘let V be a vector space over F and let e_1,e_2,…,e_n be a basis for V’. I think the exams did not involve any computation and maybe involved producing some of the proofs from the course or applying them to something.