Assignments 4

1. Find the best approximation of
   $f(x)=e^x$ by a quadratic polynomial, with respect to the inner product
   
   $$⟨f,g⟩={\int }_{-1}^{1}f(x)g(x)dx.$$
2. Exercise 1.9 at p.125.
3. Exercise 2.1 at p.128.
4. Exercise 3.10 at p.134.
5. (Prove it if the statement is true or give a counter example otherwise)
   Let
   $L:V\to W$ be an isomorphism,
   $V$ and
   $W$ are inner product space. Suppose
   $\{v_1,\cdots ,v_n\}$ is an orthogonal basis of
   $V$, then
   $\{L(v_1),\cdots ,L(v_n)\}$ is also an orthogonal basis of
   $W$.

• Assignment4_solution