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Q8E

Expert-verifiedFound in: Page 1

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**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.**

**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.

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 \(w = - {\bf{u}} + 2{\bf{v}}\).

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 \(x = - 2{\bf{u}} + 2{\bf{v}}\).

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 \(y = - 2{\bf{u}} + 3.5{\bf{v}}\).

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.

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