System of linear equations:

Values of and :

Linear Combination:

We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically.

In this module we will focus on vectors in and .

Express as a linear combination of and . Interpret your results geometrically.

We
need to find scalars and such that
This amounts to solving a system of linear equations

Use your favorite method to solve this system. (Hint: adding the second equation to the first will work well for this system.) You will find that and . Now we can write as a linear combination of and as follows:

Geometrically speaking, this means that the vector is the diagonal of the parallelogram determined by and . The original vectors and are shown below together with the parallelogram and its diagonal.

Use geometry to express as a linear combination of and .

In addition to the Explanation below, the following video illustrates this process.

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We are looking for to be the diagonal of a parallelogram determined by scalar
multiples of and .

Because a scalar multiple of a vector can point in the same direction as the vector or in the opposite direction, we will start by drawing straight lines determined by the two vectors.

The two lines that we drew will contain the sides of the parallelogram we are looking for. To find the other two sides we will draw lines parallel to and through the head of vector .

Now the parallelogram is clearly visible.

The last remaining task is to identify the sides of the parallelogram as scalar multiples of and . We do this by identifying vectors and as the vectors that determine the parallelogram.

Observe that vector is half the length of and points in the opposite direction, while the vector is the same length as and also points in the opposite direction.

Now we can write as a linear combination of and as follows

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.

Given two non-collinear vectors and in , and a vector which is not a scalar
multiple of either or , we can express as a linear combination of and as
follows:

- (a)
- Draw lines and determined by and , respectively.
- (b)
- Through the head of vector , draw lines and , parallel to and , respectively.
- (c)
- Let be the point of intersection of and .
- (d)
- Let be the point of intersection of and .
- (e)
- Let denote the origin. Then and for some scalars and .
- (f)
- We have .

Can the vector be written as a linear combination of vectors and ?

We will start
with a geometric approach.

Observe that and are scalar multiples of each other and lie on the same line.

A linear combination of and has the form:

This shows that all linear combinations of and will be scalar multiples of , and therefore lie on the same line as . Since does not lie on the line determined by it cannot be expressed as a linear combination of and .

We can also address this question algebraically. To express as a linear combination of and , we need to solve the equation. This gives us a system of equations

When you try to solve this system using your favorite method, you will find that the system is inconsistent. Thus, cannot be written as a linear combination of and .

Set up a system of equations that can be used to express as a linear combination of ,
, and , or to determine that such a combination does not exist. Do not solve the
system.

We are looking for , , and such that
This translates into the following system of equations:

At this point you may not have the techniques for solving this system of equations efficiently. So, we present this problem as an example of a set-up only. Later in the course you will learn to solve and interpret such systems.

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

System of linear equations:

Values of and :

Linear Combination: