/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Let \(P\) be the transition matr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 45

Let \(P\) be the transition matrix from \(B^{\prime \prime}\) to \(B^{\prime},\) and let \(Q\) be the transition matrix from \(B^{\prime}\) to \(B\). What is the transition matrix from \(B^{\prime \prime}\) to \(B ?\)

Short Answer

Expert verified
The transition matrix from \(B^{\prime \prime}\) to \(B\) is the product of the two given transition matrices, i.e., \(P \cdot Q\).

Step by step solution

01

Understand and set up the problem

We need to find the transition matrix from \(B^{\prime \prime}\) to \(B\). To do this, we need to use the transition matrices \(P\) and \(Q\) that we're given. Remember, \(P\) is the transition matrix from \(B^{\prime \prime}\) to \(B^{\prime}\), and \(Q\) is the transition matrix from \(B^{\prime}\) to \(B\). The transition matrix we're looking for essentially represents the transition from \(B^{\prime \prime}\) to \(B^{\prime}\) (this done by \(P\)), and then from \(B^{\prime}\) to \(B\) (done by \(Q\)). So to find it, we need to multiply \(P\) and \(Q\).
02

Perform the matrix multiplication

Now we need to perform the matrix multiplication \(P \cdot Q\). To do this, remember that we multiply the entries in the rows of the first matrix with the entries in the columns of the second one and sum them up. The result is a new matrix, which represents the combined transitions.
03

Conclusion

The product of the matrices \(P\cdot Q\) is the transition matrix from \(B^{\prime \prime}\) to \(B\). This is because it contains the combined transition information of the two steps: \(B^{\prime \prime}\) to \(B^{\prime}\) and \(B^{\prime}\) to \(B\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the transition matrix from \(B\) to \(B^{\prime}\) by hand $$\begin{array}{l}B=\\{(1,0,0),(0,1,0),(0,0,1)\\} \\\B^{\prime}=\\{(1,0,0),(0,2,8),(6,0,12)\\}\end{array}$$

Use a graphing utility or computer software program with matrix capabilities to find the transition matrix from \(B\) to \(B^{\prime}\) $$\begin{array}{l}B=\\{(1,3,3),(1,5,6),(1,4,5)\\} \\\B^{\prime}=\\{(1,0,0),(0,1,0),(0,0,1)\\}\end{array}$$

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime}+y^{\prime}=0 & \\{1, \sin x, \cos x\\} \end{array}$$

Prove that the set \(\left\\{e^{a x}, e^{b x}\right\\}\) is linearly independent if and only if \(a \neq b\)

Use a graphing utility with matrix capabilitics to (a) find the transition matrix from \(B\) to \(B^{\prime},\) (b) find the transition matrix from \(B^{\prime}\) to \(B\), (c) verify that the two transition matrices are inverses of one another, and (d) find [x] \(_{B}\) when provided with \([\mathbf{x}]_{B^{*}}\) $$\begin{array}{l} B=\\{(1,3,4),(2,-5,2),(-4,2,-6)\\} \\\ B^{\prime}=\\{(1,2,-2),(4,1,-4),(-2,5,8)\\} \\\ {[\mathbf{x}]_{B^{\prime}}=\left[\begin{array}{r} -1 \\ 0 \\ 2 \end{array}\right]} \end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.