Okay, so let's now solve few questions which are related to rank. So let's go to this question where I am saying that rank of A and rank of A slash B, which means the augmented matrix are not equal. Then what you can infer from this? So let's understand this.
Okay, I mean let's understand this that what is the relation between rank of A and rank of AV. Let's suppose that this is A, this is A. And then, You bring one more vector that you are saying b and then you are saying that okay there is something called as ab.
So let me write ab here only. So you say that this is a but you bring one more vector and then you are writing that vector as b. Now when you are bringing one more vector then what are the possibilities?
I mean let's suppose in a you have three linearly independent vectors. Suppose okay. This 3 is just a sub number that I am giving an example.
And now suppose that you bring one more vector which is b. So what's going to happen if you bring one more vector which is b? Just think about it. Either this b is linearly dependent on these 3 vectors or this is linearly independent on these 3 vectors.
I mean linear combination of these 3 vectors or b is completely new vector. So if b is completely new vector, then in that case what will happen? In that case, you will say that bringing this b is adding you. Okay, see, let me go to the next page.
What I mean to say, I am not sure if you are understanding here or not. So, let's just go to the next page. So, what I mean to say is that you will bring one more vector b. In this A, if A is having three linearly independent vectors.
Now, suppose you bring one more vector b. So, this b is either linear combination of these three vectors or it is not a linear combination of these three vectors. If it is a linear combination of these three vectors, then you are not adding anything, which means you can say rank of A equal to rank of AB if you are not adding anything.
If this B is actually a linear combination of these vectors. So see, that's very nice thing that this is actually telling you that B is a linear combination. Okay, LC means linear combination.
I hope that's fine. So LC means linear combination or let me write. capital L. So if you are saying that rank of A and rank of AB are equal, then it must mean that B is a linear combination of vectors in A, right?
So that's very nice thing. And now suppose that both are not equal, which means rank of A is not equal to rank of AB. Then in that case, what do you infer?
In this case, you will say, if both ranks are not equal, it means that B is completely new vector. See, this is the complete matrix AB. Now, if these are not equal, then what does this mean? It means that you have added something, right? Okay, let me again go to the next page.
See, what I mean to say, there is one vector that you are going to add. Now, this vector can add the rank by plus 1 or nothing. This vector is b and this is a matrix. So what I am saying here is that this vector whenever you are bringing can add a rank to plus 1 or it won't add anything.
Which means that rank of ab is either rank of a or rank of ab equal to 1 plus rank of a. Which means it will at most differ by 1. Because there is one more vector that you are bringing in, either that vector is basically already included in A, which means this is already linear combination of vectors of A, or if this vector is new, then it will add a rank to 1. This is going to happen, right? Now, here, in the second case, where if these are not equal, then it means what? It means that rank of A is basically, or rank of AB is basically 1 plus C. If they are not equal, it means it.
this B is actually has added something then it means what it definitely means or vice versa also it definitely means that B is not a linear combination of columns of A okay so this is very important to understand that if both ranks are equal then B has not added anything if both ranks are not equal then actually B has added one vector I mean one linearly independent vector which means b is not a linear combination of columns away and interestingly let me go to the next page interestingly you can infer something from here See, this is A, this is B. Now, there are two cases. One is that rank of A equal to rank of AB. Another case is that rank of A is not equal to rank of AB. So, this not equal is actually 1 plus rank of A, okay, and then equal.
I mean, either you write not equal or you write 1 plus rank of A and then equal. Both are same. Now, here you can say, that b is a linear combination of columns columns in a or columns of a okay b is a linear combination of columns of a and here you can say that b is not a linear combination of columns of a right and now if b is a linear combination of columns of a It means that solution exists.
Now, you currently don't know whether the solution is infinite or finite. But yeah, solution exists. If both ranks are equal, it means that solution exists. Which means that whatever you are bringing in, that is actually a linear combination of columns of A.
And I can produce that vector. Then solution exists. If whatever you are bringing in is a new vector, then solution does not exist, right? Then solution does not exist.
exist. So that's how you can work with the whole story. I hope you understood. Now let's just try to understand the same thing, the same this thing with the zero non-zero concept. Okay, which means that suppose this is a, this is a and then you bring one more vector which is b.
Now let's just see that what's going to happen. If I say rank of A is equal to rank of AB. Can I say here that 0, 0, 0, 0, non-zero does not exist or something like that.
Can I say does not exist. See, I can say it. Why?
Just think about it. If both ranks are equal, it means what? I mean, you can maybe try to prove with the backward reasoning. or backward reasoning or like let's just say it proof by contradiction which means that suppose suppose this exists which means if zero zero zero zero zero zero non-zero exists somewhere non-zero exists somewhere then this complete matrix will be having one extra pivot right see what is happening here that let me just go to the next page that you are having one matrix which is a and there are some pivots in a right let's suppose this is the pivot maybe there is no pivot here which means that you can say that okay whatever it is i mean this is star star star and then maybe there is no pivot here but let's suppose there is a pivot here and let's suppose there is no pivot in this column also which means this could be anything but then this is 0 0 okay suppose suppose suppose this is the case okay which means there are two pivots in a i don't know what are these stars maybe 0 maybe non-zero there are two pivots in it that's what i know now let's just tried bringing a b okay let's just now try bringing a b if you bring a b here then in that case i mean you obviously you must have brought it before uh this uh row equivalent reduction uh or the gaussian animation but i'm just trying to explain in other way okay so suppose you're bringing the b here in that case what's going to happen this b could look like something this okay uh if both ranks are equal then it means what it should not add any pivot Which means that here, after this line, after this line, all must be 0. After this line, okay, I mean, b is just one vector, okay. So, here you are just going to add single, single numbers, okay, single, single numbers.
So, after this line, these two numbers must be 0, isn't it? Or instead of extra, I mean, multiple 0 rows, let me just work with one 0 row. So, basically what I am saying here is that you bring one more b.
In that case, this must be 0. It cannot be a pivot element because ranks are equal. So, this must be 0. See, it will depend on this particular element. Let's suppose I am currently writing it anything.
Okay, maybe some cloud or something. So, basically, maybe let me write it like this. Okay, suppose that this particular element is 0. Then in that case, what will happen? And if this is non-zero, then in that case what will happen if it is non-zero it means it is a pivot If it is 0, then it means both ranks are equal, right? If it is 0, then both ranks are equal.
If it is non-zero, then it will work as a pivot. Pivot in the sense that for the complete matrix, if you talk about the rank, so for the complete matrix, it will be a pivot element. So if it is non-zero, then rank, it will increase the rank basically.
So rank of AB equal to 1 plus rank of A. If it is non-zero, if it is 0, then both are equal. Okay, so see things are very simple.
This was a whatever a you take, okay, whatever a you take, if you add one more vector, if that vector is actually adding one, one pivot, it means that in some if it is see if that's that is the case, it means it must be adding the pivot. If both ranks are not equal, it means it must be adding the pivot. It means what if it is adding a pivot, it means what that there must be a row where like where in A it is not having any pivot but in AB it is having one pivot right see this rank of AB is not equal to rank of A let's just try understanding that what does this mean see this is basically what I am saying you bring one more vector okay if both are not equal it must mean that this B is actually adding one extra pivot to this complete matrix, isn't it?
Extra pivot adding means what? That there was no pivot in this particular row. There was no pivot at all.
And then suddenly you have a pivot here, right? If both ranks are not equal, then it means that you have one pivot here. Let me know if that is clear.
If both ranks are not equal, then it must mean that earlier you were not having any pivot, which means all are 0, all are 0. And then, then suddenly you got a pivot, right? Now, if both are equal, then what will happen? If both are equal, then earlier also you were not having any pivot, okay? Earlier also, you were not having any pivot in some row.
And now you are adding. It is not adding any pivot, which means that it is 0 only. It is 0 only. So basically, the final summary is that, if both ranks are equal, then it means that this B is a linear combination of columns of A. Because this B is not adding anything.
Then it must mean that B is a... linear combination linear combination of columns of a right it it it is same as saying so or from here you can directly say it is same as saying that 0 0 0 0 non-zero non-zero does not exist And from here also you can say directly, if B is a linear combination of columns of A, then if all are zeros, and since it is a linear combination, then this non-zero cannot come here, right? Because linear combination of zeros cannot produce you non-zero, okay?
And similarly, you can say that if both are not equal, then B is not a linear combination, okay? B is not a linear combination of columns of A. It means 0 0 0 if it is not a linear combination of columns of A, it means that 0 0 0 non-zero exist. I mean Gaussian animation will make sure that 0 0 0 non-zero exist. If B is not a linear combination of columns of A.
I mean see it is not necessary in general like you have any vector B and if it is not a linear combination of columns of A, then you have 0 0 0 non-zero. I mean, it is not necessary in general. But after Gaussian elimination, you will find some row like this.
Okay. I mean, before the Gaussian elimination, it might not be true, obviously. I mean, we are talking about after the Gaussian elimination.
So, this thing is after the Gaussian elimination. Okay. Right.
So, 0, 0, 0, non-zero, this row I am talking about after the Gaussian elimination. So, now I hope this thing is clear that this B is adding something or B is not adding something. Now, let's just answer this question that if both ranks are not equal, it means B is adding something.
B is some extra thing, then no solution, right? Okay, let's just do more questions probably. So here no solution, but we can do more questions, right? If both are equal, then let me know what is the answer. It means maybe unique solution, maybe infinite solution.
It means that B is a linear combination of columns of A. That's the only thing you can infer from here. Maybe unique, maybe infinite, right? I hope you understood.
So that is the story I wanted to talk about. Now, let's just see the earlier flowchart we had. So 0 0 0 non-zero exist and then we say the inconsistent solution.
If it does not exist, then we said, okay, if there is a free variable, then infinite solution or otherwise unique solution. Can we have the similar flowchart for the rank? Let's just see.
So see, if both ranks are not equal, it means B is adding something. It means inconsistent solution, right? Or both ranks are not equal. It means it also means it is just one-to-one mapping. If 0 0 0 non-zero exist, it means that both ranks are not equal.
So you think about the matrix in terms of after the Gaussian animation only. Okay. If 0, 0, 0, non-zero exists, it means what?
There is one extra pivot you got. It means that both ranks are not equal. So this is same as saying both ranks are not equal.
And this does not exist. It means that if there is a 0 earlier and then here also it is 0. It means that both ranks are equal actually. Right.
Both ranks are equal. And then after that, you will just check. Okay.
I mean earlier you were just checking free variable or not but after that you will just check whether both ranks are equal that is okay but now you have now you have rank that is equal to n which means every column has pivot so rank equal to n means every column has pivot every column has pivot if every column has pivot means what all the columns are linearly independent I mean no column is dependent on anything else you there is no no free no free variable right which means if there is no free variable then uh if i say there is no free variable then it means what it means that all the columns are having the pivot if all the columns are having the pivot then rank must be number of the pivot elements which are number of the columns which is n right so rank must be n don't worry we will solve the questions and after that it will be more clear so see this free variable and the rank are basically very much related If I want to convey this information that free variable does not exist, then I convey this information using the rank. How? I will say rank of A is n. It means all columns are private column.
Because n is the number of columns and all columns are private column. So I will say, if I want to convey that free variable does not exist, then I will say all columns are private column. I will say every column has private, which means rank is actually n.
If I want to convey this information that free variable actually exists, Then I will say that okay rank is less than n. Right. Rank is less than n. So I am in the case where rank are already equal. Then I am talking about whether rank is equal to n or less than n.
So basically it is one to one mapping. Which means if you want to convey this information that free variable exist or not. You can convey this information using the rank.
Okay. Using the rank. Just do one thing.
I just request you to you know pause this video or maybe once this video is complete. Then just reiterate that. why it is one-to-one mapping and so on okay just just think about it because it takes some time initially now there are few questions that i will solve in the next video and i will leave these questions to you so one two three four these are the four questions i will i will put these four questions in the annotated notes without the solution and then i will be solving in the next video okay