Pauls Online Notes : Linear Algebra

Paul's Online Math Notes

Linear Algebra

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Subspaces	E-Book Chapter Section	Linear Independence

Span

In this section we will cover a topic that we’ll see off and on over the course of this chapter. Let’s start off by going back to part (b) of Example 4 from the previous section. In that example we saw that the null space of the given matrix consisted of all the vectors of the form

We would like a more compact way of stating this result and by the end of this section we’ll have that.

Let’s first revisit an idea that we saw quite some time ago. In the section on Matrix Arithmetic we looked at linear combinations of matrices and columns of matrices. We can also talk about linear combinations of vectors.

Definition 1 We say the vector w from the vector space V is a linear combination of the vectors ,all from V, if there are scalars so that w can be written

So, we can see that the null space we were looking at above is in fact all the linear combinations of the vector (7,1). It may seem strange to talk about linear combinations of a single vector since that is really scalar multiplication, but we can think of it as that if we need to.

The null space above was not the first time that we’ve seen linear combinations of vectors however. When we were looking at Euclidean n-space we introduced these things called the standard basis vectors. The standard basis vectors for were defined as,

We saw that we could take any vector from and write it as,

Or, in other words, we could write u and a linear combination of the standard basis vectors, . We will be revisiting this idea again in a couple of sections, but the point here is simply that we’ve seen linear combinations of vectors prior to us actually discussing them here.

Let’s take a look at an example or two.

Example 1 Determine if the vector is a linear combination of the two given vectors.

(a) Is a linear combination of and ? [Solution]

(b) Is a linear combination of and ? [Solution]

(c) Is a linear combination of and ? [Solution]

Solution

(a) Is a linear combination of and ?

In each of these cases we’ll need to set up and solve the following equation,

Then set coefficients equal to arrive at the following system of equations,

If the system is consistent (i.e. has at least one solution then w is a linear combination of the two vectors. If there is no solution then w is not a linear combination of the two vectors.

We’ll leave it to you to verify that the solution to this system is and . Therefore, w is a linear combination of and and we can write .

[Return to Problems]

(b) Is a linear combination of and ?

For this part we’ll need to the same kind of thing so here is the system.

The solution to this system is,

This means w is linear combination of and . However, unlike the previous part there are literally an infinite number of ways in which we can write the linear combination. So, any of the following combinations would work for instance.

There are of course many more. There are just a few of the possibilities.

[Return to Problems]

(c) Is a linear combination of and ?

Here is the system we’ll need to solve for this part.

This system does not have a solution and so w is not a linear combination of and .

[Return to Problems]