9.E: Exercises for Chapter 9

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

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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