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Chapter 1: Q2E (page 1)

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

Short Answer

Expert verified

The vectors are \(u + v = \left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right]\) and \(u - 2v = \left[ {\begin{array}{*{20}{c}}{ - 1}\\4\end{array}} \right]\).

Step by step solution

01

Write the condition to add the two vectors

From the given vectors, it is observed that both vectors contain two entries. So, the vectors can be denoted as \({\mathbb{R}^2}\).

Add the corresponding terms of \(u\)and \(v\)to compute \(u + v\).

02

Compute the sum of the vectors

Obtain vector \(u + v\)by using vectors \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\),and \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

\(\begin{aligned}{c}u + v &= \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{3 + \left( 2 \right)}\\{2 + \left( { - 1} \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{3 + 2}\\{2 - 1}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right]\end{aligned}\)

Thus, the vector is \(u + v = \left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right]\).

03

Write the condition for scalar multiple of a vector by a constant

Multiply each entry of a \({\mathbb{R}^2}\) vector by a scalar number to obtain the scalar multiple of a vector by a constant.

04

Compute the vector

Vector \(u - 2v\)can be written as\(u + \left( { - 2} \right)v\).

Obtain the scalar multiple of vector \(v\)by scalar \(\left( { - 2} \right)\), and then add the resultant vector with vector \(u\).

\(\begin{aligned}{c}u + \left( { - 2} \right)v &= \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right] + \left( { - 2} \right)\left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{\left( { - 2} \right)\left( 2 \right)}\\{\left( { - 2} \right)\left( { - 1} \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 4}\\2\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{3 - 4}\\{2 + 2}\end{array}} \right]\end{aligned}\)

Solve further to get:

\(u + \left( { - 2} \right)v = \left[ {\begin{array}{*{20}{c}}{ - 1}\\4\end{array}} \right]\)

Thus, \(u - 2v = \left[ {\begin{array}{*{20}{c}}{ - 1}\\4\end{array}} \right]\).

Therefore, \(u + v = \left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right]\), and \(u - 2v = \left[ {\begin{array}{*{20}{c}}{ - 1}\\4\end{array}} \right]\).

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Most popular questions from this chapter

A large apartment building is to be built using modular construction techiniques. The arrangement of apartment on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three bedroom units, and 8 one bedroom units. 7 two bedroom units and 8 one bedroom units. Each floor of plan B includes 4 three bedroom units, 4 two bedroom units, and 8 one bedroom units. Each floor of plan C includes 5 three bedroom units, 3 two bedroom units, and 9 one bedroom units. Suppose the building contains a total of \({x_{\bf{1}}}\) floors of plan A, \({x_2}\) floor of plans B, and \({x_{\bf{3}}}\) floors of plan C.

a. What interpretation can be given to the vector \({x_{\bf{1}}}\left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\\{\bf{8}}\end{aligned}} \right)\)?

b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.

c. (M) Is it possible to design the building with exactly 66 three bedroom units, 74 two bedrooms units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form.

5. \(\begin{aligned}{c}{x_1} + 3{x_2} + {x_3} = 0\\ - 4{x_1} - 9{x_2} + 2{x_3} = 0\\ - 3{x_2} - 6{x_3} = 0\end{aligned}\)

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linearly independent set. Justify each answer.

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3. \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 2}\\{ - 2}&1&6\\3&{ - 2}&{ - 5}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{ - 1}\\7\\{ - 3}\end{array}} \right]\)
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