9.E: Exercises for Chapter 9

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$Schilling.jpg$
Contributed by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling
Professor (Mathematics) at University of California, Davis

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Calculational Exercises

1. Let $ (e_1 , e_2 , e_3) $ be the canonical basis of $ \mathbb{R^3} $ , and deﬁne
\[ f_1 = e_1 + e_2 + e_3, ~~~~~~~~~f_2 = e_2 + e_3, ~~~~~~~~~f_3 = e_3 . \]
(a) Apply the Gram-Schmidt process to the basis $ (f_1 , f_2 , f_3) $.
(b) What do you obtain if you instead applied the Gram-Schmidt process to the basis $ (f_3 , f_2 , f_1) $?

2. Let $ C[−\pi, \pi] = \{f : [−\pi, \pi] \rightarrow R \mid f ~\rm{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

\[ \inner{f}{g} = \int_{-\pi}^{\pi} f(x)g(x)dx, ~\rm{for~every} ~ f,g \in C[-\pi,\pi]. \]

Then, given any positive integer $ n \in \mathbb{Z_+} $, verify that the set of vectors

\[ \left\{ \frac{1}{\sqrt{2\pi}},\frac{sin(x)}{\sqrt{\pi}}, \frac{sin(2x)}{\sqrt{\pi}}, \ldots , \frac{sin(nx)}{\sqrt{\pi}}, \frac{cos(x)}{\sqrt{\pi}}, \frac{cos(2x)}{\sqrt{\pi}}, \ldots, \frac{cos(nx)}{\sqrt{\pi}} \right\} \] is orthonormal.

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

\[ \inner{f}{g} = \int_{0}^{1} f(x)g(x)dx, ~\rm{for~every} ~ f,g \in \mathbb{R_2}[x] . \]

Apply the Gram-Schmidt procedure to the standard basis $\{1, x, x^2 \} $ for $\mathbb{R_2}[x] $ in order to produce an orthonormal basis for $\mathbb{R_2}[x]$ .

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 $(v_1 , v_2 , v_3 ) $ of $\mathbb{R^3}$ , and call the resulting orthonormal basis $(u_1 , u_2, u_3)$.

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 $P$ .

6. Give an orthonormal basis for $null(T )$, where $T \in \cal L(\mathbb{C^4} ) $ is the map with canonical matrix

\[ \left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 &1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} \right) \]

Proof-Writing Exercises

1. Let $V$ be a ﬁnite-dimensional inner product space over $ \mathbb{F}$. Given any vectors $u, v \in V$ , prove that the following two statements are equivalent:

$ (a) \inner{u}{v} = 0 $

$(b) \norm{u} \leq \norm{u + \alpha v}$ for every $\alpha \in \mathbb{F} $.

2. Let $n \in \mathbb{Z_+}$ be a positive integer, and let $a_1 , \ldots , a_n , b_1 , \ldots , b_n \in \mathbb{R} $ be any collection of $2n$ real numbers. Prove that

\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n ka_{k}^2 \right) \left( \sum_{k=1}^n \frac{b_k^2}{k} \right) \]

3. Prove or disprove the following claim:
Claim. There is an inner product $ \inner{\cdot}{\cdot}$ on $\mathbb{R^2}$ whose associated norm $\norm{\cdot}$ is given by the formula
\[ \norm{(x_1 , x_2 )} = |x_1| + |x_2 | \]
for every vector $ (x_1 , x_2 ) \in \mathbb{R^2}$ , where $ | \cdot | $ denotes the absolute value function on $ \mathbb{R} $.

4. Let $V$ be a ﬁnite-dimensional inner product space over $\mathbb{R}$. Given $ u, v \in V $, prove that
\[ \inner{u}{v} = \frac{ \norm{u+v}^2 - \norm{u-v}^2}{ 4}\]

5. Let $V$ be a ﬁnite-dimensional inner product space over $\mathbb{C}$. Given $ u, v \in V$ , prove that

\[ \inner{u}{v} = \frac{ \norm{u+v}^2 - \norm{u-v}^2}{ 4} + \frac{ \norm{u+iv}^2 - \norm{u-iv}^2}{ 4}i.\]

6. Let V be a ﬁnite-dimensional inner product space over $\mathbb{F}$, and let $U$ be a subspace of $V$. Prove that the orthogonal complement $U^\perp$ of $U$ with respect to the inner product $ \inner{\cdot}{\cdot}$ on $V$ satisﬁes

\[ dim(U^\perp ) = dim(V ) − dim(U).\]

7. Let $V$ be a ﬁnite-dimensional inner product space over $\mathbb{F}$, and let $U$ be a subspace of $V$. Prove that $U = V$ if and only if the orthogonal complement $U^\perp$ of $U$ with respect to the inner product $\inner{\cdot}{\cdot}$ on $V$ satisﬁes $U^\perp = \{0\}$.

8. Let $V$ be a ﬁnite-dimensional inner product space over $\mathbb{F}$, and suppose that $P \in \cal{L}(V) $ is a linear operator on $V$ having the following two properties:

(a) Given any vector $v \in V , P (P(v)) = P (v)$. I.e., $P^2 = P$.

(b) Given any vector $u \in null(P) $ and any vector $v \in range(P ), \inner{u}{v} = 0$.

Prove that $P$ is an orthogonal projection.

9. Prove or give a counterexample: For any $n \geq 1$ and $A \in \mathbb{C}^{n \times n} $, one has

\[null(A) = (range(A))^\perp .\]

10. Prove or give a counterexample: The Gram-Schmidt process applied to an an orthonormal list of vectors reproduces that list unchanged.

Contributors

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