/*! 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 42 Write \(\mathbf{v}\) as a linear... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 42

Write \(\mathbf{v}\) as a linear combination of \(\mathbf{u}\) and \(\mathbf{w},\) if possible, where \(\mathbf{u}=(1,2)\) and \(\mathbf{w}=(1,-1)\). $$\mathbf{v}=(1,-1)$$

Short Answer

Expert verified
The vector \(\mathbf{v}\) can indeed be written as a linear combination of vectors \(\mathbf{u}\) and \(\mathbf{w}\). The coefficients for the linear combination are \(c = 0\) and \(d = 1\). Therefore \(\mathbf{v} = d \mathbf{w} = \mathbf{w}\).

Step by step solution

01

Definition of Linear Combination

A vector \(\mathbf{v}\) can be expressed as a linear combination \(\mathbf{v} = c \mathbf{u} + d \mathbf{w}\) of vectors \(\mathbf{u}\) and \(\mathbf{w}\). The goal is to find the coefficients \(c\) and \(d\).
02

Setup System of Equations

Because \(\mathbf{v} = c \mathbf{u} + d \mathbf{w}\) this can be written as a system of equations, as the equalities need to hold for each corresponding component of vectors \(\mathbf{u}, \mathbf{w}, \) and \( \mathbf{v}\). The system is: \(1 = c + d\) and \(-1=2c - d\).
03

Solve the System of Equations

Solving this system of equations finding \(c\) and \(d\) such that both equations hold true. By adding both equations, one gets : \(0 = 3c\), hence \(c = 0\). Substituting \(c = 0\) into the first equation, it follows that \(d = 1\). So, \(\mathbf{v} = c \mathbf{u} + d \mathbf{w} = 0 \mathbf{u} + 1 \mathbf{w}\).

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 coordinate matrix of \(X\) relative to the standard basis in \(M_{3,1}\) $$X=\left[\begin{array}{r}1 \\ 0 \\ -4\end{array}\right]$$

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 \([\mathbf{x}]_{B^{*}}\) $$\begin{array}{l} B=\\{(4,2,-4),(6,-5,-6),(2,-1,8)\\}, \\\ B^{\prime}=\\{(1,0,4),(4,2,8),(2,5,-2)\\} \\\ {[\mathbf{x}]_{B^{\prime}}=\left[\begin{array}{r} 1 \\ -1 \\ 2 \end{array}\right]} \end{array}$$

Is the scalar multiple of a solution of a nonhomogeneous linear differential equation also a solution? Explain your answer.

Find the coordinate matrix of \(p\) relative to the standard basis in \(P_{2}\) $$p=-2 x^{2}+5 x+1$$

Identify and sketch the graph. $$4 y^{2}+4 x^{2}-24 x+35=0$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

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.