Chapter 9, Inner product spaces Video Solutions, Linear Algebra

Problem 1

Let $\left(e_{1}, e_{2}, e_{3}\right)$ be the canonical basis of $\mathbb{R}^{3}$, and define
$$
f_{1}=e_{1}+e_{2}+e_{3}, \quad f_{2}=e_{2}+e_{3}, \quad f_{3}=e_{3} .
$$
(a) Apply the Gram-Schmidt process to the basis $\left(f_{1}, f_{2}, f_{3}\right)$.
(b) What do you obtain if you instead applied the Gram-Schmidt process to the basis $\left(f_{3}, f_{2}, f_{1}\right) ?$

Wendi Zhao

Numerade Educator

01:24

Problem 2

Let $C[-\pi, \pi]=\{f:[-\pi, \pi] \rightarrow R \mid f$ is continuous $\}$ denote the inner product space of continuous real-valued functions defined on the interval $[-\pi, \pi] \subset R$, with inner product given by
$$
\langle f, g\rangle=\int_{-\pi}^{\pi} f(x) g(x) d x, \text { for every } \mathrm{f}, \mathrm{g} \in \mathrm{C}[-\pi, \pi]
$$
Then, given any positive integer $n \in \mathbb{Z}_{+}$, verify that the set of vectors
$$
\left\{\frac{1}{\sqrt{2 \pi}}, \frac{\sin (x)}{\sqrt{\pi}}, \frac{\sin (2 x)}{\sqrt{\pi}}, \ldots, \frac{\sin (n x)}{\sqrt{\pi}}, \frac{\cos (x)}{\sqrt{\pi}}, \frac{\cos (2 x)}{\sqrt{\pi}}, \ldots, \frac{\cos (n x)}{\sqrt{\pi}}\right\}
$$
is orthonormal.

Subhadeepta Sahoo

Numerade Educator

03:49

Problem 3

Let $\mathbb{R}_{2}[x]$ denote the inner product space of polynomials over $\mathbb{R}$ having degree at most two, with inner product given by
$$
\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x, \text { for every } \mathrm{f}, \mathrm{g} \in \mathbb{R}_{2}[\mathrm{x}]
$$
Apply the Gram-Schmidt procedure to the standard basis $\left\{1, x, x^{2}\right\}$ for $\mathbb{R}_{2}[x]$ in order to produce an orthonormal basis for $\mathbb{R}_{2}[x]$

Wendi Zhao

Numerade Educator

02:02

Problem 4

Let $v_{1}, v_{2}, v_{3} \in \mathbb{R}^{3}$ be given by $v_{1}=(1,2,1), v_{2}=(1,-2,1)$, and $v_{3}=(1,2,-1)$. Apply the Gram-Schmidt procedure to the basis $\left(v_{1}, v_{2}, v_{3}\right)$ of $\mathbb{R}^{3}$, and call the resulting orthonormal basis $\left(u_{1}, u_{2}, u_{3}\right)$.

Wendi Zhao

Numerade Educator

04:01

Problem 5

Let $P \subset \mathbb{R}^{3}$ be the plane containing 0 perpendicular to the vector $(1,1,1)$. Using the standard norm, calculate the distance of the point $(1,2,3)$ to $P$.

Eric Mockensturm

Numerade Educator

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Problem 6

Give an orthonormal basis for $\operatorname{null}(T)$, where $T \in \mathcal{L}\left(\mathbb{C}^{4}\right)$ is the map with canonical matrix
$$
\left(\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1
\end{array}\right)
$$

Nick Johnson

Numerade Educator