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Question

If a $3 	imes 8$ matrix $A$ has rank $3,$ find $\operatorname{dim} \mathrm{Nul} A, \operatorname{dim} \operatorname{Row} A$ and $\operatorname{rank} A^{T} .$

Subham Jyoti Mishra · Accepted Answer

and the question. We have a 3 to 8 metrics on its given that, but a B three interviews metrics the bank. I&#x27;ll be It&#x27;s given us tree. Now we have to find out the diamonds off. No space off here in diamonds and off. Little please. Off a plants with rank off a transport. Uh huh. This is a Peru and minus diamonds and off Thio in Venice. Mhm. This is a pure, um there any number at all? Just m a number of problems crossed in image number, the rules and an example of condoms forgiven metrics. The dimension off now face a plus. Rank off is equal number off columns. So here it is. Newt Ministry is his good filed the diamonds and off, especially Ecuador diamonds, you know, problem Space air. Which is he called the rain Here. Physical tree rankled dear ones. Waas. There&#x27;s the You&#x27;re quoting Frank. It&#x27;s just equal diamonds. Show Garden&#x27;s basically transpose she&#x27;s sick. Or diamonds or not. That&#x27;s all space off here. Physical

Bobby Barnes · Answer

eso We&#x27;re giving this 6.3 metrics and we&#x27;re told the rank of it is three. So we want to figure out the dimension of the knoll of a first. There&#x27;s actually a nice little rule for this that says so. If we&#x27;re given some in by in Matrix, then rank a plus dimension of the knoll. The dimension off the know of A is equal to in So in this case and is three we know our rank is three. So then that tells us our dimension of our no space is just going to be zero. So this is a dimension of no space of A is equal to zero. So this over here is going to be zero now to get the dimension of the row space. This is going to be the same thing as our rank, because if we actually think about it, I don&#x27;t remember if you actually say this in the book or not, but I&#x27;ll kind of draw it out, and we&#x27;ll also kind of come to the same conclusion. But yeah, this should also be three. But let&#x27;s draw. But this six by three matrix might look like so six by three is rose by columns. So we have six, um, Rose here with three columns here and each place where this reading Lee Lai intersect each other will be a position. Okay. And so if we have ranked three, that means we have three pivot positions like this. And remember, Rome is really just talking about our row is really just saying, Well, what happens about the combinations of these rows here, and if we were to completely reduce it, notice everything below these since he&#x27;s become pivot positions, these air all just going to end up being zeros here early so we can reduce them down to zeros. And so because of that, these would be our only non zero rose after we go through and reduce. So the dimension of this is going to be three as well. Now to get the rink. Um, I don&#x27;t remember if they say this in the book either. But rank of a transpose is going to be the same thing as the rank of A which is going to be three. And if they didn&#x27;t say that in there, we could actually just take the transpose of this really quickly so if we were to transpose this so this would go from six by 3 to 3 by six. So now the green are going to be the rooms and the red. We&#x27;re going to be our columns. And so since this is 1122 and 33 those also go across these here. And so because of that, that means we would have to have three pivot positions in this matrix as well. So that&#x27;s how you could kind of, like, draw it out if the book doesn&#x27;t say that. But I&#x27;m pretty sure it does say it either, but yeah. So this is how we can come to the conclusion that the null space dimension zero the row space dimension is three and the rank of a transpose is also going to be three.

Cyrielle Lorio · Answer

Okay, So what we have here is a two by two matrix for in were asked to find there Oh, excellent form and to determine its trunk. So if we are the review our lessons and grow Actually what we want is to only have one, not zero and three points in the first room which is called the people. And then for us to have a zero on the remaining rose So we can do this paper for me and syriza operations, including the scaling and root adding and multiple of one room to the other and swapping toe prose. So for this example, what we&#x27;re going to do first is to perform a series of operations the road to so we will tell Drop roll toe. We want help off our one. So our solve the equation is to one and then one minus when half of school and negative 35 us went up one. So the resulting made drinks iss 21 And we know that this will cancel out and this is equal to negative seven over. They look it here they roll Negative seven. So this is already in its role. Petulant form Since the first. We only have 190 interpret in the first row and then we have zero on the second row. So were also asked to determine its round and we know. But that by definition it is the maximum number of lean Ernie independent factors in any tricks. So we are to look at our matrix. We have to non zero role Row one and row toe has non zero np&#x27;s. Therefore our ground is the

Cyrielle Lorio · Answer

Okay, So what we have here is a two by two matrix where we were asked to find that a western inform and to the 13 it&#x27;s round. So we are to review our lessons and row echelon. What they want to do here is to convert the final role to zero. So we only have one non zero entry point which is called the People. So this can be done by doing serious off operations, including re scaling a room, adding multiple of one row to the other room are swapping toe rose. So for this example, what we want to do first is to swap are one with our two. So if we are to do this, we&#x27;re going to get they got the poor e and to negative port. So it&#x27;s mentioned. We want the last rule to have zero value, so we don&#x27;t. But we don&#x27;t want any non zero Palace of the final. So we need to do another operation, which is to add our two. We&#x27;d 1/2 so far one So our resulting matrix is that I&#x27;m sorry. Negative for eight and then to plus when have a negative four. The negative four less. One have off it. So this will be fancy out. Steal it. This will be zero and our resulting matrix. Is they getting four e 00 So this is now there. Oh, excellent form off the meat. Thanks. So we want to know the round. And based on definition rockets, the maximum number of leaner and independent better send me pics. So since this is a two by two matrix, the excellent form has one non zero, which is the first role. Therefore, they&#x27;re drunk, is what?

Cyrielle Lorio · Answer

Okay, So what we have here is supported by board meat. Thanks. When we&#x27;re asked the fine, it&#x27;s reduced row, excellent form. And it&#x27;s round. So based from our previous nous, since we need to convert disperse with its throw at you. And for that I mean it&#x27;s strong and no further operations in Britain introduced row for So this your instrument for muses deceive elementary row operations We throw incident for but this different great beer. Yeah, So we say that, uh, Matrix has seen its reduced ruined student for if it&#x27;s stopping spice the four requirements first, the 1st 90 number in the first row is number. What set one? The second row also starts with the number one, which is further to the right than the leading enter in the first row. So for every subsequent roll, the number one must be further to the right, so it should look like this. Resembling as thirties. It&#x27;s so three, the leading Andrian each room must be the only non zero in its Kahlo. And finally, and in an zero, those are placed at the bottom six. So we can do this by going serious. A parliamentary row operations, including in their changing wonder with another multiplying one drove and non zero Pakistan and replacing one room with one row plus a constant times another rope. So for this example, what we&#x27;re going to do first is to transform it into its role. Excellent For so our first step is to swap Are one is this what are one? We&#x27;d our port. So by doing that, we know have for negative too. 30 eight, three negatives, too. 07 Cool. Negative one Oh, four and one negative. One negative one. Next they want to cancel the leading efficient in art. And we do this by doing arto. Sorry, we do this by doing are too minus three ports so far. What? By doing that, we know how for negative too. But he eight zero They get the one house negative 9/4 one. So negative one Oh, four and one negative One negative one. Next we want to cancel. The leading coefficients are three and we do this by doing our three minus one help off our one With that, we now have four Negative too. 30 eight zero negative one house negative. 9/4 one zero zero one have zero and finally one negative one Negative. What? For our fourth step, I have your stand following. What we want to do now is to cancel the leading coefficient in our port. So we do this by building are for minus 14 Fire went now we have for blanketing too. Three e zero negative. 1/2 negative. Mine over four and one 00 1/2 zero and then zero. Then get the 1/2 negative. 7/4, then zero. Next again, we want to cancel the leading coefficient off our port, which is negative Would happen. So what we&#x27;re going to do now is are for bind us are too. And with that, we now have four negative too. 30 eight zero. Get the 1/2 negative. Nine Gilbert four one 00 1/2 zero and then zero zero one. Have they get the one our six is to and then cancel the leading coefficient in row four. So by cancelling that peace will satisfy one off art criteria in row insurance form. So we do this now by are four by doing art for minus are the Now we have for negative too Pretty eight zero. They got the 1/2. They get the 9/4. One zero zero went help. Zero, then 000 Negative. What? So this is already on its row? Answered on form. And what we want to do now is to get drunk. So by definition, the run is the maximum number off linearly independent, better in a matrix. So this is equal to the number off vans era rose in its euro insolent meetings. So apparent a look at the example. We have 12 and three rows with non zero element, which is roll one road boom and there are three. Therefore, our room it&#x27;s three. So we&#x27;re not quite done yet since we need to get the reduce row petulant form off our matrix. So what they want to do now first is to is to divine are one we&#x27;d for. So now we get one Megat the 1/2 but imports zero negative 1/2 negative nine bill before one 00 one house zero and then 000 negative. What next? We want to me zeros in column toe except at the entry of pro to with that What we&#x27;re going to do is to Centra are who you are. What we now have one zero 30 one zero negative 1/2 Leggett The night Cooper for one zero zero. When have zero then 00 zero negative. What? So we&#x27;re almost done with our solution. So I hope yourself follow we Our next step is to multiply. Art do well, Negative that we know have one zero 30 one zero one nine over to and negative do 00 1/2 zero and then 00 zero negative. What? So for our next there, What we want now is to make zeros in column three and step at the entry of three. So we don&#x27;t this by subtracting six are three who are one with death. We now have 1001 zero one 9/2 negatives do. 00 1/2 zero 000 Negative. What? Oops. X over and now this is Jay. This is I&#x27;m just gonna rewrite businesses. Step que who? Not those Any confusion. So we&#x27;re now at step ill and what we&#x27;re going to go now is to said nine rt Who are you? So we now have 1001 zero, 10 Negative too. 00 1/2 0000 What? Or step em. What we want to do now is the multiply our city. We thought so We get one 001 010 Negative 00 10 then 00 zero Negative four step and we&#x27;re almost done. What we want to do now is to make zeros in column for accepting the entire approach. For So we do this by Oops, I&#x27;m being are one. And I report No, we have 1000 010 Negative, too. 0010 and 000 Negative. So far. Step. Oh, we&#x27;re going to set it up now. Two off are for so I do. Now we have one 000 010 The room 0010 000 Negative. What? And for our final step, what we&#x27;re going to do now is to multiply our for with negative one. And what we&#x27;re going to get is 1000 0100 0010 and 000 What? So this is now that reduce role? Excellent form off the medics

Cyrielle Lorio · Answer

Okay, So what we have here recently before matrix where it were asked to get its reduced her alleged in four minutes. Trump. So based from our previous ness sense, we need to convert this first with its your alleged inform to determine its trump and no further operations to convert it to reduce row excrement for so destro incident for uses the same bro operations. We grow excellent form, but that&#x27;s different criteria. So we say that a matrix is in its reduced row. Excellent for if it&#x27;s not despise the for informants first, the first nuns it&#x27;ll number in the first of all is the number one second. The second row also starts with the number one, which is further to the right than the leading and a versatile So for every subsequent through the number one must be further to the right, so it should look like this resembling a stir kicks turn. The living entry in each row must be the only and then zero number in its column and finally and in an zeros our place at the bottom of the takes So we can do this by performing several elementary role operations include the interchange of wonder with another multiplying one drove a Nancy Drew constant and replacing one throw with one drop, plus a constant times another rope. So for this example, we&#x27;re going to transform it first to its role. Excellent For so for our first step, what we&#x27;re going to do is to swap roll one windrow Oops, Windrow three. Sorry. By doing that, we now have for negative eight, three, then three Negative six, 27 one negative to one eight. Next, you want to cancel the leading coefficient off our toe by performing? I do minus three forts off our one. By doing the computation on your own, you should get four negative. Eight Ready, then zero zero negative one over for. And that could be 1/2 one negative, too. One and three. And for our final step, we want to cancel the leading coefficient of R three. So we do this by doing arty minus one fort. So far, one we know have for negative eight, three and 00 negative. 1/4 and negative one has 00 14 and want help. So this is no the role and should inform off The Matrix were also asked to determine gets drunk and by definition, the wrong is the maximum number of linearly independent Becker in its matrix. So this is equal to the number of non zero rose in its row, actually, for so in our example, we have 12 and three rows with non zero element, which is throw one row two and three. Therefore, all were ground history. So we&#x27;re not done yet since we&#x27;re asked to get its produce row insolent for so for hours. For what we want to do now is to Dubai are one with four. Okay, now we have one negative school, but he over for in five paths 00 negative 1/4 and negative 1/2 and 00 1/4 and one hats since the element of proto and call them to equal zeros. We need to swap those. But first we need to find the first non zero element in the column toe. Under the people entry, a scandal seen. There is no such entry. So move to the next room. We want to make zeros in column three, except at their entry, a troll toe fallen three. With that, our next step is to on 30. Thanks. Art who or what? We that we now have one Negatives too. Zero one zero zero negative. 1/4. And they get the 1/2 in the 00 one over for in one hand for a step. A step. If what we want to do now is to add are pretty and start with that. We now have one negative too. Zero and one zero zero negative. 1/4 and they get the 1/2 and then 0000 for our Step G, which is our final step. We want to multiply. I do. We negative for to satisfy of Liberia. Our final answer. Then it&#x27;s one negative too. 01 001 and 0000 So this is no our final answer and they reduce row. Excellent form off our meetings.

Problem

If a $6 \times 3$ matrix $A$ has rank $3,$ find $…

Problem 5 Easy Difficulty

If a $3 \times 8$ matrix $A$ has rank $3,$ find $\operatorname{dim} \mathrm{Nul} A, \operatorname{dim} \operatorname{Row} A$ and $\operatorname{rank} A^{T} .$

Answer

3

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