CHAPTER 02.19: VECTORS: Rank a set of vectors Example 2 In this example, we’ll look at what the rank of a
set of vectors is. SO let’s suppose somebody says: given A1 vector equal to
25, 64, 89; then another vector is given as A2 is equal to 5, 8, and 13. And another
vector is given as A3 is equal to 1, 1, 2. Find the
rank of the set of vectors. So we are doing that – the first thing which we
know is that the rank of the set of vectors is going to be less than or equal
to three. Because the dimension of the vectors is three, so it has to be – it
can only be less than or equal to three. Let’s see what the number is which
is less than or equal to three. We already know from a previous example that
A1, A2, and A3 are linearly dependent. So A1, A2, and A3 are not linearly
independent: they are linearly dependent. So that means that your rank of a
set of vectors, which is given to you, the three vectors which are given to
you; they’re going to be less than or equal to 2, because it cannot be 3 -
because you don’t have 3 vectors which are linearly independent. So how do we
figure out –hey, is it 2, is it 1, is it 0? If you look at A1 and A2, let’s
suppose, if I Take A1 and A2 they are linearly independent. You can show
that. So you can show that A1 and A1 are linearly independent, then in that
case the rank of set of vectors is going to be equal to 2. And that’s the end
of this segment. |