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Linear Algebra and its Applications
Found in: Page 1
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

8. Vectors w, x, y, and z

The vectors are \(w = - {\bf{u}} + 2{\bf{v}}\), \(x = - 2{\bf{u}} + 2{\bf{v}}\), \(y = - 2{\bf{u}} + 3.5{\bf{v}}\), and \(z = - 3{\bf{u}} + 4{\bf{v}}\). Yes, any vector in \({\mathbb{R}^2}\) can be expressed as a linear combination of u and v.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Steps to write vectors as a linear combination

u and v are given along with the coordinate axis shown in the given figure.

To write the vectors as a linear combination of u and v, the first step should start from the origin and move in the direction of the desired vector. Then, to reach toward the desired vector, a number of steps should be used.

Step 2: Steps to reach vector w

The following are the steps required to reach vector w as shown below:

  • From the origin, move 1 unit in the negative direction of u.
  • Then, move two steps in the direction of v.

In the vector form, it is represented as \(w = - {\bf{u}} + 2{\bf{v}}\).

Step 3: Steps to reach vector x

The following are the steps required to reach vector x as shown below:

  • From the origin, move 2 units in the negative direction of u.
  • Then, move two steps in the direction of v.

In the vector form, it is represented as \(x = - 2{\bf{u}} + 2{\bf{v}}\).

Step 4: Steps to reach vector y

The following are the steps required to reach vector y as shown below:

  • From the origin, move 2 units in the negative direction of u.
  • Then, move 3.5 steps in the direction of v.

In the vector form, it is represented as \(y = - 2{\bf{u}} + 3.5{\bf{v}}\).

Step 5: Steps to reach vector z

The following are the steps required to reach vector z as shown below:

  • From the origin, move 3 units in the negative direction of u.
  • Then, move four steps in the direction of v.

In the vector form, it is represented as \(z = - 3{\bf{u}} + 4{\bf{v}}\).

Thus, the obtained results are \(w = - {\bf{u}} + 2{\bf{v}}\), \(x = - 2{\bf{u}} + 2{\bf{v}}\), \(y = - 2{\bf{u}} + 3.5{\bf{v}}\), and \(z = - 3{\bf{u}} + 4{\bf{v}}\).

The figure implies that any vector in \({\mathbb{R}^2}\) can be expressed as a linear combination of u and v because the grid can be stretched in every direction.

Most popular questions for Math Textbooks

Suppose A is an \(n \times n\) matrix with the property that the equation \(A{\mathop{\rm x}\nolimits} = 0\) has at least one solution for each b in \({\mathbb{R}^n}\). Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.

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