
# 9.E: Exercises for Chapter 9

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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.

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