Quiz 3

• p.124: 1.1, 1.2, 1.3, 1.4, 1.7
• Suppose
  $V$ is an inner product space,
  $W$ is a vector space and
  $T:V\to W$ is an isomorphism. For
  $w_1,w_2\in W$, define
  
  $$⟨w_1,w_2⟩:=⟨T^{-1}{w}_1,T^{-1}{w}_2⟩,$$
  where the right-hand side involves the given inner product on
  $V$. Prove that this defines an inner product on
  $W$.
• Show that, for real
  $\mathbf{u}$ and
  $\mathbf{v}$,
  
  $$⟨\mathbf{u}+\mathbf{v},\mathbf{u}-\mathbf{v}⟩=\Vert \mathbf{u}{\Vert }^2-\Vert \mathbf{v}{\Vert }^2.$$
• Suppose
  $V$ is an inner product space,
  $⟨\cdot ,\cdot {⟩}_1$ and
  $⟨\cdot ,\cdot {⟩}_2$ are both inner products defined on
  $V$. Show that
  
  $$⟨\mathbf{u},\mathbf{v}⟩=⟨\mathbf{u},\mathbf{v}{⟩}_1+⟨\mathbf{u},\mathbf{v}{⟩}_2$$
  defines an inner product on
  $V$.

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Quiz 04/10

• quiz0410