Linear Combinations of Vectors

When studying vectors, the two main operations we have learned about are vector addition and scalar multiplication. Both are involved in the important concept of a linear combination of vectors.

For example, is a linear combination of , and because

In this section we will focus on vectors in and .

Visualizing Linear Combinations in and

Answer the questions below using the GeoGebra interactive. To use the interactive, you can
(a)
Change vectors and by dragging the tips of these vectors.
(b)
Change the coefficients and of the linear combination by using sliders.

(a)
Let and . Find and such that
(b)
Let and . Find and such that
(c)
Use the same vectors and as in the previous part. Do you think it is possible to express any vector in as a linear combination of and ? Yes, No
(d)
Let and . Do you think it is possible to express any vector in as a linear combination of and ? Yes, No
Visualizing linear combinations of vectors in is more difficult than doing so in . The following GeoGebra interactive will help you do this.
To use the interactive, define vectors , and . Use sliders to change the coefficients and of the linear combination. The linear combination is shown as the pink vector along the diagonal of the parallelepiped.

RIGHT-CLICK and DRAG the left panel to rotate the image.

Geometry of Linear Combinations

The method we used in Example ex:lincombparallelogrammethod to express the given vector as a linear combination of two other vectors is sufficiently useful that we summarize the steps.

This GeoGebra interactive will allow you to go through the steps given in Procedure pro:lincombgeo for a combination of vectors of your choice. To use the interactive
(a)
Enter components of vectors and .
(b)
Enter components of vector that you want to express as a linear combination of and .
(c)
Use the navigation bar to go through the steps of Procedure pro:lincombgeo

From Geometry to Algebra of Linear Combinations

One of the stipulations in Procedure pro:lincombgeo is that vectors and should be non-collinear. You can use the interactive in Exploration exp:proc4 to investigate what happens when and are collinear. The following example examines what happens from a geometric as well as an algebraic standpoint.

Practice Problems

Solve a system of linear equations to express as a linear combination of and .

System of linear equations:

Values of and :

Linear Combination:

Use Procedure pro:lincombgeo to express as a linear combination of and .

Linear Combination:

Use two different approaches (algebraic and geometric) to explain why the vector cannot be expressed as a linear combination of vectors and .
We have seen Procedure pro:lincombgeo applied to vectors in . The same process can, in certain cases be applied to vectors in . In both parts of this problem you will be asked to follow the steps in Procedure pro:lincombgeo to express one vector as a linear combination of two given vectors. Then you will be asked to identify the condition which makes it possible to do so.
The following GeoGebra interactive shows vectors , and . RIGHT-CLICK and DRAG to rotate the image.

(a)
Can be expressed as a linear combination of and ?
No, because is not between and . Yes, because all three vectors are in the same plane. Yes, because all three vectors are in the same plane, AND and are not collinear.
(b)
Use the navigation bar at the bottom of the interactive window to view construction steps of Procedure pro:lincombgeo applied to vectors , and . (Right-click and drag to rotate the image.) Use the final image to express as a linear combination of (blue) and (red).
The following GeoGebra interactive shows vectors , , and . RIGHT-CLICK and DRAG to rotate the image. Use geometry to explain why cannot be expressed as a linear combination of and .

We can also show that is not a linear combination of and algebraically by attempting to solve a system of equations corresponding to Set up the system of equations

Find the reduced row echelon form.