Hello there. So for the following exercise, we need thio way Have some matrices and we need thio Calculate the rank off the matrix on the nullity off the metrics on. Well, that's that. The information off these two, basically we got some metrics. And what the serum say is that Suppose that a is in our three? So is a matrix off sm matrix three by three on the reels. Okay, so basically the the serum say that the rank off a plus deniability off the metrics, A should be close to three Onda. Uh, well, we need to to test that. Or which is the same that the rank off the metrics a plus, the nullity off the metrics. A should be close to the number off columns. Uh, off the metrics. A Okay, so well, we're going to put, uh, this on the test. So we're going Going to just basic. So let's start. Okay. So for the first metrics, um, we got the following one minus 13 five minus four minus 476 on to Okay, So this is our matrix. So how to compute the rank off the matrix? Basically what we need to do is to reduce the Matrix to the Asian form by performing some role operations. I'm going to do the full procedure for this the first metrics. But for the following, I am just going to put their reduced form. Because otherwise this video will be really, really long. Okay, so with this exercise for this, I'm going to show the whole procedure. So first, basically, the Asian form means that we need to put zero here. You're here. You're here on put once here. And here's how to do that. Well, we just need Thio. Make some raw operations on. You will see. That is not that complicated. So I the first step, I'm going to put zeros on this part on this part here on seven. On the five I try to I will try to put a zero. Okay, so the second row I'm going to put the operation here will become will be the second row minus five times the first roll on for the third role. What I'm going to do is multiplying. Well, basically, will be our three. Here is our story. So here will be Are the third row minus seven times the first road. With that, you will see that we're going to paint a one on zero A zero here and here. Okay, So the film role remained the same on the second one Started changed 01 minus 19 and hear the same. Okay, so here you can notice something. So these two roles are the same. So basically, in the next step, I'm going to put zero on this whole row. How to do that? Well, we just need to subtract. So the operation is that are is there are three thorough will become the thorough minus the second row. Technically, so here we have taken 01 minus 19. And here's 000 So this is the Asian form, the reduce it nation form of the Matrix. Here, you can see that we need for the rank. Okay, So the rank off this matrix a will be the number off rose on the reduced action form of Dimitri's. A So, Or the number off the Yes. He is just a number off overalls that we got here. The number off Bibles, Basically, that's the real definition. So here the rank is just because we only have two pilots here, here and here. So the rank off these metrics is equals to do just by using the theory, Um, we can attain the nullity off the system. However, we way. Actually, that's what we want to show that the, uh the theory is true. So we need to make the whole compilation. So how to compute the knowledge? So let's go to the community or other called other people. Call it the colonel off the metrics. So basically, we got we need to solve the system. A X equals to the zero vector. So our matrix, um, you're going Thio considered to reduce it form means these metrics here. So, uh, here we got that one more step, actually. So our vector is one minus 1301 minus 19 000 We can put a zero on this place here. So how to do that? Well, we need just to subtract this deep the heroes. So the first role will become the first Rob lost the second row. So this is 10 minus 16, 01 minus 19 00 Just to make it easier to calculations. So there's going to be or matrix a on we need to a dimension before us off this system. But this is equivalent to help the following So x one minus 16 3 on X two minus 19 x three is equals to zero on here is equal to zero. So we need to find some nontrivial solution on uh, well, for the nontrivial solution, we just we have some free variable that in this case we can 233 x x three is a is a free variable s o x three. I'm going to change it as t I'm going to represent it as just another constant, some constant t. Okay, So by doing that, the solution becomes x one. Is it close to 16? T on X two is equals to 1980. So the vector a solution for the system is given by the following its 16 de 1990. And we say that ex Christie So this corresponds to our X that the solution for this system here on this is the the The solution on as you can see, is just one vector. You cannot separate this. So the solution X is just We can represent it in this way. 16. 19 1 on. He's just made by one vector. Okay, so is any scale. Basically, you can put anybody here. So the newly space, the newly space eyes, a span by only one vector that is 16, 19 and one. Okay, so the whole null space it generated by scaling scaling vectors off this form. So it's just much buying by any tea. Here. This'll vector on the obtained, the solution for the north space. So the new space eyes generated by only one vector. So the Newell nullity off the matrix A is equals to one. Yeah, Onda, we got all that we need for to prove the the theory. Um, the chick, the theory, Um so the rank off a plus, the nullity off the matrix a should be equals 23 And that is true because the rank is a constitute on the nullity is equals to one. So that is three. So that's correct. Okay, so this is the basic procedure Thio off in the role of the rank on the ability off the vector. So for the next, I'm not going to make the whole procedure to reducing the matrix because could be too long for it could take us too long. Okay, so let's continue. The next matrix that we got is the following 20 minus one. Here is for zero minus 2000 Okay, so here we got some, uh, we see that We've got something here. Okay? So, actually, you can observe that these matrix, this role here is to times these the first role. So the operation that we need to do is that the second role will be equal. We're going to make an operation here, so we'll be, um, two times the first role, minus the second role. And by doing this, I'm going to put a zero on this place. But actually, what I'm going to do is put in the zero to the whole row. So at the end, we only have these matrix so clearly here we only have one role under on the Asian form of the Matrix. So the rank is just one. So the rank off a is equal to one. No. Now, let's continue to the nullity. Okay? So we need to technically just solve the system. A X equals to the zero vector A will be the reduced form. So it's 20 minus 10000 times x one x two x three equals to the zero vector. Okay on. Well, the solution is given in terms. Will is just X one is equals to one half off. Three. Okay on. Well, x two, it doesn't could be any value. Okay? Yeah, ex. You could be any value. I'm going to put it equals to t. And actually x three am going to put it equals to it because it is also free variable. So here we got the solution for the new space for these systems solution for the system. So the solution is given in this way. So is the vector. Ah, that is one half s t s. Okay, so look, we can separate this vector here. So is equals to t 010 plus s one half 01 Okay, So what does mean is that the new space, the new space off the matrix A is a span by the vectors. 010 Um, one half 01 The whole most space is generated by linear combinations off these two vectors. That's what the meaning off this that this T and s could be any value could be any in the real value is that here? We got a linear combination off two vectors, and these two vectors generate the whole new space. So the nobility is just the number off the generators off in the space. So the nullity off the metrics A is equals to two because we got two vectors here. Or you can see it as we got to Victor's here as a linear combination. This here. We cannot make this DISIP oration on the previous example because you can see that t is multiplying the whole vector. We cannot separate. Okay, so the in this case, the spaces just generated by only one factor. However, here you can see that is generated as two vectors because you need the linear combination off to off these factors. So in this case, the reality is equals to two. So we got that the rank off the matrix is equals to one. The nullity is equals to to, so we can check that the rank off a plus, the nullity off a is equals to three. Because this disease is, uh, yeah, this rank here is equals to one. This note it is constituted is equal to three. So they're equal. So we have checked that this you satisfied for these metrics? Let's continue. Let's see the next matrix. So the next matrix is 1452 2130 minus 13 to 2. Okay, In this case, note that the column vector the number of columns is four. So, basically, in this case, we need to check that the rank off this matrix plus the nullity off the matrix should be equals to four. Because in this case, these vectors are on our four. Okay, so we need to satisfy duck. The period examples is a matrix that have three columns. So we need to check this case are four. So the theory and say that we need to check that the round personality should be equal to four. Okay, so let's continue. So, after reducing these metrics to the Asian reform, I'm not going to put the procedure, protect too much time. So after making some row operations here, we obtain the following Asian reduced form off the matrix here. Zero want want here for divided seven? I hear 0000 Okay, so here the rank is equals. Due to we got only two roles on the reducing the Asian for so the rank off the metrics a easy constitute. That's the first thing. The second part is competing minority. So, as I mentioned before, and as we have done before, we need to find a solution for this system. The Matrix A is Chris correspond to this one. So what we've got here is 101 minus two. Divide seven 0114 The way. The seven on dhere 0000 that multiply by x one x two x three x four and this should be equal to 0000 Okay, so the trade find solution for the system. So here for so X one is equal to so the first. Okay, so here we got three equations on forum knowns. Actually, we got only two equations here. Two equations on the forum known. So we're going to obtain to free variables. Okay, Those free variables will corresponds to x three and x four because it doesn't have any pilot on that on the metro. Okay, so x three and x four are going to be your free variables. You can You're going to to observe the for example, in the X one. The X one is, of course, to minus 63 plus two divided seven x four on the solution for X two is equal to minus 63 minus four divided seven x four. Okay, so the solution is given in terms off X three and x four, so we could put any value to them. Also, X three could be s on X four is going to be t. So after doing this, the solution The solution for the system in this case X is equals to minus s plus two divided 70 minus s minus four, divided 70. And here s anti. Yeah, so you may observe that here we can separate this. So we on one side we had we got s that is equals to minus one minus 110 on T on the other side. So here we got two divided seven here, minus four, divided 70 and one. So, in this case, the newly space is generated by linear combinations off these two vectors, as I expressed before, basically What I'm saying is that the newly space off a is a span by linear combinations of these two vectors, minus one minus 110 on the vector two divided seven minus four. Divided 701 So the nullity off this vector, so at the end, nullity off the victory. A easy constitute on us. Uh, as as we expected, the rank off a plus, the nullity off A This is equals to two plus two. This is equals to four. And we expected to have tea before. Okay, because it should be equal to the number of columns. The number of columns off the matrix A on that. Okay, so that is, uh, true. We have check. Let's continue. Yeah. So the next matrix is the following one for 569 three minus two one for minus one. My number 10 minus one minus two minus one, 235 78 Okay, so this is the next matrix in this case. Note that the number of columns is equal to five. So in this case, we need to check that the rank plus the nullity should be equals to five. So this is what we need to check. So just remember that the number of columns is equal to five. Okay, so here again, we need to reduce the Matrix to the Asian form. So after performing some raw operations, we obtain the following matrix 10 1 to 1 01111 0000000 Okay, so, after reducing the metrics to the actual informed within the following on Dhere, you can observe that we got only two rows. So in this case, the rank off the matrix is equals, Tutut. Now we need to check The nullity toe again is a X equals to the zero vector in this case is equivalent to say, 1012101111 00000 the mhm, X one x two x three x four X five equals 20000 Okay. Okay. So look. Uh huh. Here's the one. Okay, so here we got only two equations. Three or known? 50 no. So, technically, three are going to be free variables, and we're going to check that so x one is equal to minus x three minus x four two times x four on minus x five The x two is up is equal to minus X 63 minus x four R minus X five and that's all. So that is the solution. So here, x three exploring an X five are free variables we can put. Actually, we need to put some constants there. So let's, uh, say that three Ys equals two s. X four is equals to t. The next five is equals to are so could be any value. Could it could be any in real value here. So if we do this, replacing this to the solution will think that X will be, um, here minus s. So here is minus s minus two team my news are minus s minus D minus. Are S D r. Okay, on you can see that in this case, we can separate this dissolution in is made by a linear combination of three factors. So what we obtain is the following s minus one minus 1100 plus T minus two minus 1010 Um, and those that are most by by our so here will obtain minus one. So here is minus one my 1001 So, in this case, the new space the responsible solutions off this system are represented at linear combinations off three vectors. 123 formerly has expressed it on the other example. On the on the other exercise, it means that the new space is a span by these vectors minus one minus 1100 plus minus two minus +1010 on plus minus one minus +1001 Okay, so it it is a span. Here is a separate commissary by cons. Okay, so here the new spaces generated by three vectors. So the new space, the reality actually, the reality off The matrix A is equal 23 Okay. And here previously, check the Iraqis equals to to so v rank off a plus. Nullity off. A should be equal to the number off columns. But the number of columns is equals to five. And here the nudity is equal to three on the wrong Waas equals Tutut. So this is true Good. It was very fine. Finally we got the last metrics. Okay, so the last metric that we need Thio do this whole procedure is a following. So here is one minus 32 to 1 036 zero minus three. Tu minus three minus 244 396065 minus two nine to minus four on minus five. So now we've got these metrics. Okay, so in this case, again, we got 35 columns. So the number of columns with this So the number of columns is equals to five. Okay. So, again, we need here to reduce these metrics to the action form. Here is for just in case. Okay. Okay. So after performing some row operations, we obtained the reduced form off. They reduce it matrix to the veteran form on it. Corresponds to do for thes matrix 100 t seven divided six 01 00 minus one. Divine six 001 zero minus five by the 12 on Dhere to rose just off zeros. Okay. Yeah, So here we just need to count the number off Rose here. We got three rows. So the rank off the metrics a easy close to three. The number off rose off the reduce natural. Now, let's compute venality, so we need to find the solutions off the system on that corresponds to 100 276 0100 minus one Divided six 0010 minus five Divine 12 00000 And here 0000 on this multiplied by x one x two x three x four x five, and this should be equal to 00000 Okay, so there's the system that we need to find a solution. So x one is equal to mhm minus two x four minus seven divided six x five x two is equal to 16 x five and finally X three is equal to five divided 12 x five. Okay, so in this case, the free variables our X four and X five So x four I'm going to write equals two s on X five is going to be equal city to the solution in this case, for the space is X. Here is minus two s minus seven divided six deep one six de five divided 12 de s anti. Okay, so in this case, we can separate this system into just two, so those that are multiplied by s or here is minus two 0010 On those multiply lights to hear you. The minus 7/6 one divided six five divided 12, 0 and one. So in this case, the new space is generated by two vectors. So as we have seen before, then the little will be just too. So the knowledge team off the metrics, A will be equals to do so the rank. So we need to verify that the rank off a toe, the nudity off a is equals to the number off columns off the metric A. Yeah, the number of columns off the matrix A is equals to five. The nobility is equals to two. And we check that the rank WAAS equals 23 So this is true on that for this exercise.