• The term test took place Tuesday, October 17, 11-1 in Examination Centre
  • No homework due this week! Homework #5 is due 11pm on Friday, October 27
  • Required reading: Chapters 2.1, 2.2
  • We covered: Linear transformations
  • Additional homework (do not hand in).
  • We didn't have enough time to discuss Lagrange interpolation in class, but this will be covered in tutorials. The upshot:

    Given distinct $c_0,\ldots,c_n\in F$, there are unique polynomials $p_0,\ldots,p_n\in Pol_n(F)$ such that $p_i(c_j)=1$ if $i=j$, $=0$ otherwise. (These Lagrange interpolation polynomials are given by explicit formulas, which you must know.)

    Theorem: The Lagrange interpolation polynomials $p_0,\ldots,p_n$ form a basis of the vector space $Pol_n(F)$.

    (Idea of proof: they are linearly independent, and there are $n+1$ of them.) Given $a_0,\ldots,a_n\in F$, there is a unique polynomial $p\in Pol_n(F)$ such that $p(c_i)=a_i$, given by the linear combination $p=a_0p_0+\ldots+a_np_n$. Make sure to practise this in concrete examples! (Note also that for finite fields, we must have $n<\# F$ in the discussion, since the $c_0,\ldots,c_n$ must be distinct.)