Double Trouble

I have come to the conclusion that although there are four academic classes listed in my Cornell student account, I’m really just taking the same class four times. [I am aware that the title of this post is “Double Trouble,” however, “Quadruple Trouble” doesn't rhyme, plus my liberal studies class is a little different. Mass and Energy Balances also has slightly different material, but I take it with all the same people as p-chem and linear algebra (to a lesser extent), so all three might as well be the same class.]

For example, we spent a good portion of the beginning of linear algebra talking about linear transformations because any linear transformation can be written as a matrix that’s multiplied by the vector or object to be transformed, and linear algebra is all about matrices. Meanwhile, in p-chem we were introduced to operators. Some operators are linear. Here’s how to tell if a transformation or operator is linear:

From linear algebra:

Definition: A linear transformation from Rⁿ to R^m is any function T(x) with two properties:

                1) T(u+v) = T(u) + T(v)

                2) T(cu) = cT(u)

From p-chem:

Linear operators

                Â[c₁f₁(x) + c₂f₂(x)] = c₁ Âf₁(x) + c₂ Âf₂(x)

Recently in linear algebra, we were thinking of functions as vectors in the context of inner products and related material.

Been there, done that.

Our linear algebra professor also made this note about inner products.

Don't forget the complex conjugate

And then last week we started our day off in physical chemistry learning about quantum spin and were introduced to the symmetrization and antisymmetrization operators. These operators have eigenvalues of 1 and -1 and work with the permutation operator to form the spin portion of the wavefunction for electrons. We went on to Mass and Energy Balances, had lunch, and ended the day in linear algebra, where the following happened:

We were writing quadratic functions as a vector x multiplied by a matrix A which was multiplied by x.  That’s not too difficult, but in order to apply a theorem we’d been working on, A needed to be symmetric.  Magically, if you call this symmetric matrix B and write B as (A+A^T)/2 + (A-A^T)/2, B will be symmetric.

The professor told us that the first matrix (A+A^T)/2 was symmetric, so naturally I made a comment to my friend about how (A-A^T)/2 should be called antisymmetric, in light of what we’d done earlier that morning in p-chem. I was completely kidding, but the next words out of the professor’s mouth were something along the lines of “and the second matrix is antisymmetric.” I’m not sure of his exact phrasing because all the ChemEs in linear algebra were too busy laughing, groaning, and/or crying in despair. ChemE is taking over our lives.