Q51E Question: - Show that dds(Sx鈫�)... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q51E (page 428)

Question: - Show that $\frac{d}{d s} (S \overset{鈫�}{x}) = S \frac{d \overset{鈫�}{x}}{d t}$ .

Short Answer

Expert verified

Answer

The solution is $\frac{d}{ds} (S \overset{鈫�}{x}) = S \frac{d \overset{鈫�}{x}}{dt}$ .

Step by step solution

Explanation of the solution

Consider $\overset{鈫�}{x} (t)$ be a differentiable curve in Rⁿ and S be an nxn matrix.

Now, consider the system as follows.

$\frac{d}{d t} (S \overset{鈫�}{x}) = \frac{d}{d t} [\begin{matrix} a_{11} & 鈰� & a_{1 n} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} & 鈰� & a_{n n} \end{matrix}] [\begin{matrix} x_{1} \\ 鈰� \\ x_{n} \end{matrix}] = \frac{d}{d t} [\begin{matrix} a_{11} x_{1} & 鈰� & a_{1 n} x_{n} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} x_{1} & 鈰� & a_{n n} x_{n} \end{matrix}] = \frac{d}{d t} [\begin{matrix} a_{11} \frac{d x_{1}}{d t} & 鈰� & a_{1 n} \frac{d x_{n}}{d t} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} \frac{d x_{1}}{d t} & 鈰� & a_{n n} \frac{d x_{n}}{d t} \end{matrix}] \frac{d}{d t} (S \overset{鈫�}{x}) = [\begin{matrix} a_{11} & 鈰� & a_{1 n} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} & 鈰� & a_{n n} \end{matrix}] [\begin{matrix} \frac{d x_{1}}{d t} \\ 鈰� \\ \frac{d x_{n}}{d t} \end{matrix}]$

Simplify further as follows.

$\frac{d}{d t} (S \overset{鈫�}{x}) = [\begin{matrix} a_{11} & 鈰� & a_{1 n} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} & 鈰� & a_{n n} \end{matrix}] [\begin{matrix} \frac{d x_{1}}{d t} \\ 鈰� \\ \frac{d x_{n}}{d t} \end{matrix}] \frac{d}{d t} (S \overset{鈫�}{x}) = S \frac{d \overset{鈫�}{x}}{d t}$

Hence the proof of the solution.

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91影视

Question: - Show that $\frac{d}{d s} (S \overset{鈫�}{x}) = S \frac{d \overset{鈫�}{x}}{d t}$ .

Short Answer

Step by step solution

Explanation of the solution

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91影视

Question: - Show that dds(Sx鈫�)=Sdx鈫�dt.

Short Answer

Step by step solution

Explanation of the solution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Geometry

Probability and Statistics

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Question: - Show that $\frac{d}{d s} (S \overset{鈫�}{x}) = S \frac{d \overset{鈫�}{x}}{d t}$ .