In this example, we're going to work with a vector space, and we're actually going to
find a basis for the vector space.
So that means any element from the vector space I can write as a linear combination
of the basis vectors.
So the set of vectors we're going to work with is this set of vectors.
The vectors 1, 0, 1, minus 2, 1, 4, and 0, 3, 1.
So that we have three vectors of length 3.
And we are asked, is this a set of vectors-- is this set of vectors a basis for R3?
So R3 is the collection of all vectors of length 3.
If this set is a basis for R3, then any vector from R3 can be written as a linear combination
of these three things.
So in this case, we're not really coming up with a basis.
We're actually asked is this provided collection of vectors a basis for R3?
So how can we figure that out?
Well, since we're working with R3, we know that we need three linearly independent vectors
because that's just the rule that we have for vector spaces.
If you have three dimensions you need three linearly independent vectors to form a basis.
So really the question becomes, given that I've been provided these three vectors, are
they linearly independent?
Are these three things linearly independent?
So what I need to do is I need to check for linear independence of the three vectors.
And there are a few different ways that we can do that.
Let's go ahead and do it the first way.
And we'll also do it-- part B we'll do it using determinants.
But in this first way, what we'll actually do is we will write down a matrix corresponding
to the three vectors, and we'll do some row operations to reduce it.
So the first row operation I'll do is E3.
Replace E3 with E3 minus E1.
So rows two and one don't change, but for row three it looks like this.
Then I can replace row three with row three minus 6E2 to get rid of that sixth layer.
So I'll have a 0, a 0, and a 1.
I can go ahead and replace E2 with E2 minus 3E3.
So rows one and three don't change, but row two will turn into 0, 1, 0.
And then I can replace row one with row one plus 2E2.
So rows two and three don't change.
So that row one, or equation one, will be 1, 0, 0.
So I've performed row operations, and I was able to turn my starting matrix-- because
each column corresponds to a vector in our basis-- I've been able to transform it into
an identity matrix.
So if I look at this, what do I see?
I see three pivots.
Here's one.
Here's two.
And here's three.
So those three pivots tell me something pretty important.
It tells me that I have three linearly independent vectors.
So it turns out that these vectors are linearly independent.
If I had had only two pivots, then I would know that just two of them were linearly independent.
But since I have three-- I have three linearly independent vectors, and this does form a
basis for R3.
Another way to work this problem is to use determinants.
So let's think about that.
We know how to use determinants to check for linear independence.
So one way to do this would be just to compute the determinant of the matrix that we form.
If we actually take that matrix that we started with-- the 3 by 3-- and compute its determinant,
you get the number minus 17.
We're not going to go through all the computations by hand to do this.
I have other videos that do that.
But computing that determinant gives us the number minus 17, which is not equal to 0.
And that's the important part.
If it was equal to 0, 0 would imply that they are linearly dependent.
Since the determinant is not equal to 0, that means that they are linearly independent.
So computing the determinant tells us that we have three linearly independent columns,
which means we have three linearly independent vectors, which means we have enough vectors
to form a basis for R3.
So this is another way that you could check that the given vectors do indeed form a basis
for R3.
Obviously, if you are working with some other vector space-- R2 or R4 or R7-- you would
need to be given two vectors or 4 vectors or 7 vectors to figure out if the collection
of vectors was linearly independent.
And you could use either the determinant method or the reduced row approach to check the number
of pivots.
Either of those methods would be a legit way to check for is the given set of vectors a
basis for the vector space.