Real inner product spaces

For each matrix $A$, determine whether the formula $\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}^{T}A\mathbf{v}$ determines an inner product on $\mathbb{R}^{2}$.

Consider the inner product space $C[0,1]$ as in Example 11.5. Compute the following inner products:

Consider the inner product space $C[0,1]$ as in Example 11.5. Compute the following norms:

For $\mathbf{u},\mathbf{v}$ vectors in $\mathbb{R}^{3}$, define the product $\mathbf{u}\ast\mathbf{v}=u_{1}v_{1}+2u_{2}v_{2}+3u_{3}v_{3}$. Show that

Hint: first show that the operation $\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}\ast\mathbf{v}$ is an inner product on $\mathbb{R}^{3}$, then use Proposition 11.13.

In $C[-1,1]$, find (a) the angle between $x$ and $x^{2}$, (b) the angle between $x$ and $x^{3}$.

. In this exercise, we will work out the details of Example 11.7. We must show that $\mathbf{Hilb}_{\mathbb{R}}$ is a vector space. We will do this by showing that it is a subspace of $\mathbf{Seq}_{\mathbb{R}}$. Further, we must show that the inner product is well-defined. This requires some knowledge of convergent series from calculus.

1. Assume $a=(a_{0},a_{1},\ldots)$ and $b=(b_{0},b_{1},\ldots)$ are square summable sequences. Show that the series $a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+\ldots$ converges to a real number. Hint: consider…