Question

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

Short answer

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 uand v.

Step-by-step solution

Steps to write vectors as a linear combination

uand vare given along with the coordinate axis shown in the given figure.

To write the vectors as a linear combination of uand 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.

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

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

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

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 uand v because the grid can be stretched in every direction.