CHAPTER 02.18: VECTORS: Rank of a set of vectors: Example 1
In this segment, we will look at an example of how to find the rank of a set of vectors. So let’s suppose somebody says “hey, given these 3 vectors: A1 is equal to 25, 64, and 144. A2 is equal to 5, 8, and 12. And A3 is given as 1, 1, and 1. Find the rank of this set of vectors.” So you are given the set of 3 vectors right here and somebody is asking for the rank of the set. In a previous example, we found out that A1 A2 A3 are linearly independent. So since we found out that, hey, A1 A2 A3 are linearly independent, so the rank of vectors-rank of set of vectors in this case will be equal to 3 because we found 3 vectors that are linearly independent. Now keep in mind you don’t simply count the number of vectors which are linearly independent but you also have to look at the dimension of the vectors. The dimensions of the vectors is 3 and as we said with the rank of the set of vectors has to always be less than or equal to the dimension of the vector, which in this case in 3. So we are alright right here.
If somebody gave you something like this: Somebody says “hey, A1 is given as 25, 64, 144, A2 is given as 5, 8, 12.” So I am using the same 3 vectors which were given to me in the example. And somebody says, hey, I’m going to give you a 4th vector. A4 is equal to 2, 7, and 19. Now in this case, whether this set of vectors is linearly independent or not all 4 of them will not matter. To decide -it will not matter to decide whether the rank of a set of vectors is 4, because the dimension of the vectors is only 3. So the only thing that you will have to show-we are going to get the rank of the set of vectors is going to be less than or equal to 3- this is because the dimension of the vector is 3. So all we have to show is whether 3 of these vectors are linearly independent.
So we already found out that if I use A1 A2 A3, they are linearly independent so that gives me that, hey, I got 3 vectors which are linearly independent. So, the maximum number of the vectors which are independent here are linearly independent is at least 3 so the rank of vectors is equal to 3. SO it doesn’t matter if it whether it forms…whether all four are independent or not. So A4 can be linearly dependent- let’s suppose on these other 3 vectors, or it can be -it can form linearly independent set of vectors- all four of them, but it won’t matter that hey it will-if you decide what the rank of the vectors is going to be less than or equal to 3. So it’s always going to be less than or equal to 3, but the reason why we say it is exactly 3 is because we have at least found 3 vectors A1, A2, A3 which are linearly independent. And that’s the end of this segment.