Quiz 4

• Given
  $n\in \mathbb{N}$, find a constant
  $C$ such that, for any
  $\mathbf{v}\in {\mathbb{R}}^n$,
  
  $$\Vert \mathbf{v}{\Vert }_1\le C\Vert \mathbf{v}{\Vert }_2.$$
• Find counter-examples for the following statements:
  • Let
    $L:V\to W$ be a linear transformation. Then
    $⟨w,L(v)⟩=0$ for all
    $v\in V$ implies
    $w=0$.
  • Let
    $L:V\to W$ be linear,
    $V$ is a vector space and
    $W$ is an inner product space. Define
    
    $$⟨u,v{⟩}_L:=⟨L(u),L(v)⟩,u,v\in V,$$
    where the right hand side involves the given inner product on
    $W$. Then the above operation,
    $⟨\cdot ,\cdot {⟩}_L$, defines an inner product on
    $V$.
• Find a polynomial
  $q(x)\in {\mathbb{P}}_2(\mathbb{R})$ such that
  
  $$p(\frac{1}{2})={\int }_0^{1}p(x)q(x)dx,$$
  for any
  $p(x)\in {\mathbb{P}}_2(\mathbb{R})$.
• Find a polynomial
  $q(x)\in {\mathbb{P}}_2(\mathbb{R})$ such that
  
  $${\int }_0^{1}p(x)\cos (\pi x)dx={\int }_0^{1}p(x)q(x)dx,$$
  for any
  $p(x)\in {\mathbb{P}}_2(\mathbb{R})$.

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Quiz 04/17

• quiz0417