/*! 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 18 Write the vector \(v\) as a line... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 18

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{r} 8 \\ 10 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] $$

Short Answer

Expert verified
\mathbf{v} = -2\mathbf{w} + 4\mathbf{u}

Step by step solution

01

Set up the Linear Combination Equation

Express the vector \(\mathbf{v}\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). This means we are looking for scalars \(\alpha\) and \(\beta\) such that: \[ \mathbf{v} = \alpha \mathbf{w} + \beta \mathbf{u}\] Substituting in the given vectors: \[ \left[ \begin{array}{r} 8 \ \ 10 \end{array} \right] = \alpha \left[ \begin{array}{r} 0 \ \ 1 \end{array} \right] + \beta \left[ \begin{array}{r} 2 \ \ 3 \end{array} \right] \]
02

Set up the System of Equations

From the equation, we obtain a system of two equations: \[ 0\alpha + 2\beta = 8 \] and \[ \alpha + 3\beta = 10 \]
03

Solve the First Equation

Solve the first equation for \beta\: \[ 2\beta = 8 \] \[ \beta = \frac{8}{2} = 4 \]
04

Substitute \beta\ into the Second Equation

Substitute \beta = 4\ into the second equation to find \alpha\: \[ \alpha + 3(4) = 10 \] \[ \alpha + 12 = 10 \] \[ \alpha = 10 - 12 = -2 \]
05

Write the Linear Combination

Now that we have found \alpha = -2\ and \beta = 4\, we can write the vector \mathbf{v}\ as: \[ \mathbf{v} = -2\mathbf{w} + 4\mathbf{u} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector
In mathematics, a vector is an ordered set of numbers that represent a point in space. For instance, the vector \( \mathbf{v} = \left[ \begin{array}{r} 8 \ \ 10 \end{array} \right] \) denotes a point in a two-dimensional space. Vectors are extremely versatile in various fields such as physics, engineering, and computer science because they can represent quantities that have both magnitude and direction.
Each component of a vector refers to its position along a particular axis. In our example, the vector \( \mathbf{v} \) shows 8 units on the x-axis and 10 units on the y-axis. This makes it straightforward to visualize and manipulate points in multi-dimensional spaces.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It is the foundation for solving systems of linear equations, like the one in our exercise.
Matrix operations such as addition, multiplication, and finding the determinant fall under linear algebra. Concepts like vector spaces and basis vectors are key to understanding how vectors can be represented and manipulated.
One common problem in linear algebra is writing a vector as a linear combination of other vectors. In our example, we express \( \mathbf{v} \) as a combination of \( \mathbf{w} \) and \( \mathbf{u} \). This involves finding scalars \( \alpha \) and \( \beta \) that satisfy the equation \( \mathbf{v} = \alpha \mathbf{w} + \beta \mathbf{u} \).
By solving the resulting system of linear equations, we can determine these scalars and understand how one vector can be constructed from others in a vector space.
System of Equations
A system of equations is a set of equations that you solve simultaneously. Each equation in the system shares variables. The solution to the system is the set of values that satisfy all equations in the system.
In the given exercise, we have the following system of equations:
\[ 0\alpha + 2\beta = 8 \] and \[ \alpha + 3\beta = 10 \]
The first equation simplifies to \( \beta = 4 \). This value is then substituted into the second equation to find \( \alpha = -2 \). By solving this system, we determine how the vectors \( \mathbf{w} \) and \( \mathbf{u} \) combine to form \( \mathbf{v} \).
This method of solving a system of equations is fundamental in both mathematics and science, helping us determine unknown variables with given constraints.

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Most popular questions from this chapter

San Diego fairy shrimp live in ponds that fill and dry many times during a year. While the ponds are dry, the [airy shrimp survive as cysts. The population is divided into three groups. Cysts that survive one dry period are in group 1 cysts that survive two dry periods are in group 2 and cysts that survive three or more dry periods are in group \(3 .\) The Leslie matrix representing the survivability and fecundity is given below. \({ }^{14}\) $$G=\left[\begin{array}{ccc} 3.6 & 0.98 & 0.65 \\ 0.5 & 0 & 0 \\ 0 & 0.5 & 0.49 \end{array}\right]$$ In the third dry period there are 8365,1095, and 310 individuals in groups 1,2, and 3, respectively. a) Find the inverse of the Leslie matrix. b) Estimate the population of each group during the second dry spell. c) Estimate the population of each group during the first dry spell.

Let \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{array}\right]\) a) Let \(B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]\). Compute AB. b) Let \(C=\left[\begin{array}{rrr}4 & -2 & 7 \\ 1 & 1 & -3 \\ -4 & 3 & 6\end{array}\right]\). Compute AC. c) Based on parts (a) and (b), what is the effect of multiplying \(\mathrm{A}\) on the left with another \(3 \times 3\) matrix? Explain why.

The number of red foxes in a habitat in year \(n\) can be modeled by $$ x_{n+1}=a\left(x_{n}+z_{n}\right) $$ where \(x_{n}\) is the number of resident foxes in year \(n, z_{n}\) is the number of immigrant foxes in year \(n\), and \(a\) is a constant. \(^{21}\) We further assume that the number of immigrant foxes \(z_{n}\) is proportional to \(M-x_{n-1}\), so that $$ z_{n}=b\left(M-x_{n-1}\right) $$ The sequence \(x_{0}=0, x_{1}=1, x_{2}=0, x_{3}=1\) \(x_{4}=0, x_{5}=1, \ldots\) alternates between 0 and \(1 .\) a) Find a second-order linear difference equation that this sequence satisfies. (Hint: A coefficient can be 0 in the recursion relation.) b) Solve the difference equation you found in part (a) with the initial conditions \(x_{0}=0\) and \(x_{1}=1\) to find a formula for the sequence that alternates between 0 and 1 .

Let \(A, B\), and \(C\) be arbitrary \(n \times n\) matrices. Explain why \(A+(-A)=0\).

A \(2 \times 2\) Leslie matrix has eigenvalues \(r_{1}\) and \(r_{2}\). Find the long-term growth rate and the long-term percentage growth rate. $$ r_{1}=0, r_{2}=1.5 $$
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