Assignments 3

1. Exercise 1.5 at p.124.
2. Suppose
   $V$ is a vector space,
   $W$ is an inner product space and
   $T:V\to W$ is linear and injective. For
   $v_1,v_2\in V$, define
   
   $$⟨v_1,v_2⟩:=⟨T{v}_1,T{v}_2⟩,$$
   where the right-hand side involves the given inner product on
   $W$. Prove that this defines an inner product on
   $V$.
3. Show that
   
   $${(a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n)}^2\le (a_1^2+2a_2^2+\cdots +n{a}_n^2)(b_1^2+\frac{b_2^2}{2}+\cdots +\frac{b_n^2}{n}),$$
   for all
   $a_1,\cdots ,a_n$,
   $b_1,\cdots ,b_n\in \mathbb{R}$,
   $n\in \mathbb{N}$.
4. Exercise 1.8( c) at p.125.
5. (The problem number is the last digit of your student ID number)
   • Use ChatGPT to find the correct answer for your assigned problem.
   • Problems.

TA solution

• Assignment3_solution