Scientific Computing Using Python - PHYS:4905 - Fall 2018
Homework #8
Due 9/25/2018


Name _______________________________________________

The answers to these questions must be hand written on paper.


1. (12) Show that the pair of conditions:

[1]   L(u + v) = L(u) + L(v)  and   L(cv) = cL(v)

which is valid for all vectors u, v and any scalar c,  is equivalent to the single condition:

[2]   L(ru + sv) = rL(u) + sL(v)

which is valid for all vectors u, v and any scalars r and s.  
Your answer should have two parts. Show that [1] implies [2], and then show that [2] implies [1].



2. (6) If Q(x²) = x³ and Q(2x²) = x⁴ is it possible that Q is a linear function from polynomials to polynomials?  Explain your reasoning.


3. (20) If f is a linear function such that $f \, \begin{pmatrix} 1 \\ 2 \end{pmatrix} = 0$ and $f \, \begin{pmatrix} 2 \\ 3 \end{pmatrix} =1$ , then what is $f \, \begin{pmatrix} x \\ y \end{pmatrix}$ ?