Chapter 14, Vector valued functions Video Solutions, Linear Algebra in Action

Problem 1

Let $p(x)=a_0+a_1 x+\cdots+a_n x^n$ be a polynomial of degree $n$ with $n$ distinct real roots $\alpha_1<\cdots<\alpha_n$, where $n \geq 2$. Show that $p^{\prime}(x)$ has $n-1$ real roots $\beta_1<\cdots<\beta_{n-1}$ such that $\alpha_j<\beta_j<\beta_{j+1}$ for $j=1, \ldots, n-1$.

Eric Mockensturm

Numerade Educator

03:13

Problem 2

Use the mean value theorem to show that if $b>a>0$, then $\sqrt{a b}-a \leq(b-a) / 2$.

Doruk Isik

Numerade Educator

03:07

Problem 3

Show that the error in approximating $(27.1)^{5 / 3}$ by $(27)^{5 / 3}$ is bigger than $3 / 2$.

Michelle Ling

Numerade Educator

Problem 4

Verify Corollary 14.9.

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02:34

Problem 5

Show that the Newton step (14.11) for solving the equation $x^2-a=0$ to find the square roots of $a>0$ is
$$
x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right) \text { if } x_n \neq 0,
$$
and calculate $x_1, x_2, x_3$ when $a=4$ and $x_0= \pm 1$.

Nick Johnson

Numerade Educator

Problem 6

Show that in the setting of Exercise 14.5
$$
\left\|x_{n+1}-x_n\right\| \leq \frac{1}{2}\left\|x_n^{-1}\right\|\left\|x_n-x_{n-1}\right\|^2 \text {. }
$$

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02:34

Problem 7

Show that the Newton step (14.11) for solving the equation $x^3-a=0$ to find the cube roots of $a$ is
$$
x_{n+1}=\frac{1}{3}\left(2 x_n+\frac{a}{x_n^2}\right) \quad \text { if } \quad x_n \neq 0,
$$
and calculate $x_1, x_2, x_3$ when $a=8$ and $x_0= \pm 1$.

Nick Johnson

Numerade Educator

01:27

Problem 8

Let $E=\{x \in \mathbb{R}: 0 \leq x \leq 1\}$ and let $f(x)=\left(1+x^2\right) / 2$. Show that:
(a) $f$ maps $E$ into $E$.
(b) There does not exist a positive constant $\gamma<1$ such that $|f(b)-f(a)| \leq \gamma|b-a|$ for every choice of $a, b \in E$.
(c) $f$ has exactly one fixed point $x_* \in E$.

Carson Merrill

Numerade Educator

01:13

Problem 9

Show that the polynomial $p(x)=1-4 x+x^2-x^3$ has at least one root in the interval $0 \leq x \leq 1$. [HINT: Use the fixed point theorem.]

Carson Merrill

Numerade Educator

Problem 10

Show that the function $\mathbf{f}(x, y)=\left[\begin{array}{c}(1-y) / 2 \\ \left(1+x^2\right) / 3\end{array}\right]$ has a fixed point inside the set of points $(x, y) \in \mathbb{R}^2: x^2+y^2 \leq 1$.

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Problem 11

Show that if $A \in \mathbb{R}^{p \times p}$ and $\left\|I_p-A\right\|<1$, then, for any choice of $\mathbf{b} \in \mathbb{R}^p$ and $\mathbf{u}_0 \in \mathbb{R}^p$, the vectors $\mathbf{u}_{n+1}=\mathbf{b}+\left(I_p-A\right) \mathbf{u}_n$ converge to a solution $\mathbf{x}$ of the equation $A \mathbf{x}=\mathbf{b}$ as $n$ tends to infinity.

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Problem 12

Show that the numbers $\varepsilon_n$ that are defined in the proof of Lemma 14.15 tend monotonically to zero as $n \uparrow \infty$.

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Problem 13

Verify the second assertion in Theorem 14.19.

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02:52

Problem 14

Verify the first assertion in Theorem 14.19 by estimating $\left\|(A+B)^n\right\|$ with the aid of the binomial theorem. [REMARK: This is not as easy as the proof furnished above, but has the advantage of being applicable in wider circumstances.]

Ankit Gupta

Numerade Educator

01:14

Problem 15

Show that if $A, B \in \mathbb{C}^{n \times n}$, then $r_\sigma(A B)=r_\sigma(B A)$, even if $A B \neq B A$.

Victor Salazar

Numerade Educator

Linear Algebra in Action

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18