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### 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

# 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.

See the step by step solution

## 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.