Covered today: We briefly reviewed matrices and determinants. This allowed us to define the Wronskian of more than two functions. We then proved Abel's Theorem for nth order linear homogeneous DEs.

For next time: 4.1 - 7,8,12,13,14,20abc (note that 20abc is just a proof of Abel's theorem when n=3, so doing this should amount to not much more than a careful re-copying of part of your notes), 21, 22, 24, plus the following problems:

1.

Prove that if a matrix A has a duplicate row, then DET(A)=0.

2.

Suppose that

A=(a_ij)

is an upper triangular matrix (that is, the i,j entry of A is zero if i>j). Prove that

$$\hbox{DET}(A) = a_{11}a_{22} \cdots a_{nn}$$

3.

Suppose that $f_0, f_1, f_2, \dots f_n$ are polynomials such that the degree of f_k is k for every $k=0,1,\dots,n$. Prove that $f_0,f_1,\dots,f_n$ are linearly independent.