in-exercises-13-and-14-assume-that-a-is-row-equivalent-to-b-find-bases-for-nul-a-and-operatornamecol

Question

In Exercises 13 and 14 , assume that $A$ is row equivalent to $B .$ Find bases for Nul $A$ and $\operatorname{Col} A .$
$$
A=\left[\begin{array}{rrrr}{-2} & {4} & {-2} & {-4} \ {2} & {-6} & {-3} & {1} \ {-3} & {8} & {2} & {-3}\end{array}ight], B=\left[\begin{array}{llll}{1} & {0} & {6} & {5} \ {0} & {2} & {5} & {3} \ {0} & {0} & {0} & {0}\end{array}ight]
$$

Doruk Isik · Accepted Answer

thing. This problem we&#x27;re going to Memphis is and now were Don&#x27;t let the medics be. Is there a little echo for actually roll echo? Horrible magics. Hey, so we&#x27;re us to find the basis for full of hay and for no, uh, you know, give it two columns, periods, columns are the basis or column A. And from this row equal, we can find the columns it is. First, call them it Hurt column And the hello said in the basic structure for Cole a would be 1213 negative by negative pipes, your fire and negative 3 to 5 Negative. These are the 1st 3rd and fifth columns. A medic. It&#x27;s a seven part B. Where to find the basis vector for No way. So we&#x27;re basically trying to find this system off for eggs equals zero. Um, using the Rocco before we can form a set of equations. And that is from 1st 1 Explain, Izzy, come through negative, too. That&#x27;s too minus for its four minus fired eggs. Now, from the second polo, we know they&#x27;re five. Fixed. Three is equally true. 764 minus eight expired. Then from the third row, we know that 99 expired is zero. So this tells us that explain the zero. And in Hubei, zero in the second and the person in question. Uh, you&#x27;re my next Want to be negative two x two Negative two x two Minus for export, eh? Extreme to be seven or 564 minus seven over five. Exports is this term is zero. So, as you can see Thanks. Do the next four are free. Really? So we can write this system as exits. Take your food X one x to extreme export and its fire, and that is equal to X to needed to 1000 plus is four negative or 07 over 51 And here and these factories are the basis for no

Subham Jyoti Mishra · Answer

does that he is. It will go. This is a four by six metrics before by five records. Who? 93. Six? Who? My When it&#x27;s too three on 3. 93 94? Uh huh. Minus six nine. Bye. Nine. When it&#x27;s too mhm. Mhm. Venice for one. Mhm. Let&#x27;s go. This mhm. This has been reduced toe. Wow. A cloned home, Pierre, which is too honest? Three 6 to 5 00 three minutes. One money, 000 We won three. Mhm. That way. So here we have to find out in the ankle. Yeah, the physical, the diamonds or Koroma. Yeah. So this is you go to the number all non zero. Uh huh. Bruise little markets. So if you have one, you go on a tree. The rank history, the diamonds, you know? No, none. Not just a physically brutal and minus drank off a Which is equal. And his number all problems, which is five minus three physical, too. Now the pace is called or loans notice here the equal then values off column vector corresponding to By what elements way have. These are the private elements, So call on one, three and four so corresponding Tothis all the metrics. We have the vectors two minus two or minus two that we have six minutes. 393 And we have proved ministry by managed for Then you can include the basis. Well, Metro north tricks. So this is really cool. The royal elements corresponding to non zero rose. So you shall be guru buh minus to be Mm. Right. Um, 00 three minutes. 11 00 zero one She So it is the basis of electric said No. This is off bottle metrics off here will be equal. So listen Vector X corresponding to see X is equal. Zero essay is the reviews for a week long form of the metrics. You see, in this case, the acquittal, the one the most three upon to 0 to 32 Little little 11 day three. I&#x27;m going through. 000 13 00000 So corresponding, Solis. And more tricks. Uh huh. Three by two x two. That&#x27;s nine by two Ex life. And we have a school here and we have minus four x i, the country for the next three. And then we have minus three x five And, uh, X bye is always no tricks. So therefore, the base he saw No mhm er the brutal he trying to hum his heroes. Heroes, heroes. No. Um Mhm nine by school zero minus four by three. Mhm minus three in the morning. Mhm. Looks like these are the Yeah.

Subham Jyoti Mishra · Answer

then we have in the role equivalent metrics off. Here it is introduced. This is a truly clone form. Yes. Yeah, In Venice, Woman Bye. Is your all minus do five minutes 6000 So here we have in our stoke beautiful their rank. Well, that hair the ranking had to say, is the group. They&#x27;re diamonds and all for no mark. Fix your hair, which is a total number. All by what columns in a Sen Roy equivalent metrics. We can see that the number of five elements our this from this. So there are two hybrid elements with the responding lee there, too. Uh, they were columns. Any. So therefore rank off is to hear now way have seminars to find out the diamonds. And so that is ankle long. Uh huh. No, not just here. So that&#x27;s sort of the core. Do Uh huh. Mm n minus. Frankel. Hey. So here and is the number all, uh, Harlem&#x27;s so here number of columns is for think of these two. This is equal to again. So the angle northern metrics Yes, to we have in us. Holds little Find out the basis. Oh, all right. Uh, huh? Room columnar Row another, which is a garland A. I can also from the markets here. That wow number of non zero. Uh huh. Rose in the store. The 1st and 2nd room on bond. The power, uh, elements are nivel stern. Second role. So the corresponding columns and here one minus one five horse toe, minus six. So this from the basis off our democracy? Yeah. Yeah. On the basis of rule, all our exterior will be the non zeros off the regular merchants. So these are one Sheetal. Then it&#x27;s one wife hand zero minus two five minus six. Wow. So this way derived from the Roy equal and metrics, Which is this? No, we have toe find out basis off model that it&#x27;s here to do that. Well, right. The reduce Rui clone Former, which is C. C will be after performing our prisons, we will have 10 minus 15 Single one. Minister of Ivory to three 0000 Yeah, you know this particularly with you know, there c 00 Looked like x one. Did you go three minus takes? Oh, and exposed will be by upon two x g plus Thanks. Very minus. Uh huh. Whoa. Mm hmm. Way have Venice, I explore. First she x four Next three. Next four. Uh, she really ever So here. Long the cases. No get. So here we look like one I bring to one Seattle. Honest by famous 301 the span off this toe vectors will give us this solution methods off. C x mhm.

Runpeng Li · Answer

all right, so in this problem work, even the matrix A and row equivalent being tricks be. So the first thing we need to do is to without calculation, to lease the run K and dimension of No A. So the run k. It&#x27;s just Ah, we just need to count on non zero rose reduced in a row, reduce toe matrix. Speak. So that is actually three nuns a year old rose. So the next thing you mentioned up, no way it&#x27;s to count. Yeah, and then about three up three barrels. So, in this case, X five, he&#x27;s always zero. If we consider the axe equal zero, the equation being very question and x $5 is always zero and ext ext. Three will be determined by export an X five but x 50 So experts three will be. We only be determined by its export, so explore it&#x27;s a pre bearable now x one will be determined by x two x three x for an X five. But on Lee, X two can be re bearable because x three will be determined by explore an X 50 so x two It&#x27;s a three variable so the dimension turns out to be too. Okay, so the next thing is to find the basis for column A or column may the basis. Uh, the only thing we need to do is to find out people people columns, and write down the people of columns in himself for entries up eight. So, by observing the matrix B, we find that people columns are come one Come three column five. So we just write down the first column there column and the fifth column off, eh? So that is one negative, too. Neck of three and three and four. Um, we have neck of six of six and four. The fifth column is nine negative. 10 and next three And there. So we&#x27;re done. Okay, next thing is to find Roadway. So in order to find the row A we still we need to observe the reduced. A rueful reduced to roll matrix, um, to observe. Dad, we just write down the rose that are not are not all zeros. So in this case, we need to exclude last row, we keep the the rest of the rose. So then we&#x27;ll be so first factor. Yes, One back up three 057 Second ISS Neural zero two, two. Sorry to negative. Three eight they call him is 0000 Okay, so the next thing is to find the basis for no way. Well, to find that now you need to do we need to do further. So we have to turn this, uh, Roy Quinlan to be relatively equivalent matrix B to, uh, reduced ancient form. So our first right down be one naked 30 by negative seven. And they&#x27;re all No. Two connective seven, a neck chemistry and eight Sorry. Hate. And they&#x27;re on their own. Zero. There are five and all deals for the last room. Okay, to do the washing elimination, where youse I say we first divide the second roll by too. So that gives 1230 by next of seven. You 01 naked, three have, and four. And we also divide by divide a 30 year old by five so that it will be the rose 0001 and all the roads for the last. Let&#x27;s roll. Okay. No. We use the second roll minus four times out four times off the third girl. So that is one negative. 30 by seven and 001 naked. Three have and zoo and last row. Sorry. 1/3 row is 00001 And all the roads for the last roll. So it needs to selling reduce station form. Okay. Um Well, the next thing is to find out, find out the free variable cleans. So that is It&#x27;s five zero on next three is we have x four and x one is real text too. Minus five x four minus seven x five well X 50 So he&#x27;s just a three x two minus five. Explore. All right, so right here we have, we have, actually, that is X three, just 3/2 off x four and x one is reacts to minus five X for Okay. So in order to find out basis, remember, we have to have to free parables here so that the sex to our next four. So the way to do that, we first have, um, check. X one will be three f x two minus by export and next to would be one zero and x three is zero x two, but three have off x four and next four is just zero and one. So our faces hey says is so by our calculation is for three. Three. Sorry, we 100 and 9 to 5. Do three have on the one. All right.

Runpeng Li · Answer

All right, so in this problem, we have, um, Matrix B. So we first need to find out a rank of eight. So the rank up way is just a non zero rose and be so that is three. And we also need to find a dimension off now. Hey, so that&#x27;s the number off the free vegetables off, eh? So that&#x27;s no A. So let&#x27;s count the number of three barrels. So first we have from the third row, we can tell that X four is determined by X five and X six. So five and six are three variables. And what&#x27;s more, from a second rule, we can tell that X two can be determined by x three x for x five x six but X for X by a nexus can be determined by because, uh I mean X value and access our free, better blacks for can be determined by these 23 variables. The only the only Rivera bill left is x x three so that sex three and X one will be determined by all the last of variables. And the rest of their bows are, uh, only three off the rest of their bows are three miracles. So this that will be three. Okay, so the second thing we need to find call a basis. So the basis in this case is just ah, we need to count the people column saying matrix B. Then we find a corresponding columns in the Matrix A and write it down. So the people columns tricks B is they are calling one to call him, too, and column four. So, in terms of columns in a ad column, one which is all wants, call him too one to an active one. Back up. Three decker, too neck of one naked three negative, too. And the fourth column. We have 7 10 1 and 9 to 5 zero, so we&#x27;re done. As for basis, upper away. Okay, the thing we need to do is to just write down the non zero rose off a trick. Speak so that is 11 negative. 37 NY Collective night and second role. 01 Negative one three. Or and give three. Okay. And the third row, which is 001 next one. And negative, too. Okay, so that&#x27;s our basis for roadway as Florida business for no a Okay, Um, we have to do the culturing elimination to find out the road. Reduce station of form. So outfits right down since the last two rows are all zero. So, uh, we can&#x27;t We don&#x27;t have to come. We don&#x27;t have to consider these two rows just because of their 1st 3 rolls. So they are 11 Next. Three, 79 Next, nine and 01 connective. 13 or negative. Three and 0001 connective one. Negative, too. Okay, so the way we do it, we first to you stuff. First thing we can we can use the first row, Um, minus the second rule. So Fuze, no. First of all, find the second rule does one minus one. That is zero naked, three minus next one. So that is connected to seven minus three. War nine minus four. Bye. Negative night. Minus negative. Three. Which is selective. I wish this positive three so that it&#x27;s negative. Six and second row for the second row. We can use the second row minus three off three times the third row. So that&#x27;s, uh so that will be the 1st 3 terms will be still be there so three minus three zero and poor minus, uh, four. Minus three times next to one. So that is positive. Three us for that is seven. And we have connective three times negative to less six minus three. So that&#x27;s three. Okay, All right. And we will keep the last roll. All right, So by our metrics, here we have, say, one to report. So asks for ISS x five last two x six by our I are less role here. And we also have x two iss hostage x three minus seven. Next five minus three x six and we have at last x one. Next one will be positive too. X three minus four x for minus five x fi plus six x six. So, as we said before, the pre variables are Xbox X six and I guess x three So you can see from our equations here x three x 563 variables or determined by five and six to depart. Determined by 3561 Determined by 3456 And yeah, right. So And the rest we need to do is just to find out defectors. We first have x three x three x five and next six five and x six All right, so for X one, we had two x three and four negative for off export. The next four is x five plus two x six. So we can do the river. Do the substitution here. So Oh, right down here. Just for more room so two x three minus four of x five plus two x six minus four start five x five Last six Stay x six. So this is two x three negative four x five minus five x five So is negative. Nine X fi a negative eight 66 So that would be plus six x six. So it&#x27;s a negative too. X six. Okay, so, yeah, so x one again. Rex one. We have two of x three and 99 off of x five the next night and negative two off exits and blacks to We have one off next three and connective seven off expire and connective three off. Six. The next three would just one zero zero And next four, we have one of x five, one of ex spy and to make six. Okay, so x five next exits are all free bearable. So X, If I would be 010 accidents will be 001 So this three vectors will be our basis or no way.

Problem

In Exercises 13 and $14,$ assume that $A$ is row …

Problem 13 Easy Difficulty

Answer

$\left\{\left[\begin{array}{c}{-2} \\ {2} \\ {-3}\end{array}\right],\left[\begin{array}{c}{4} \\ {-6} \\ {8}\end{array}\right]\right\}$

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$\left\{\left[\begin{array}{c}{-2} \\ {2} \\ {-3}\end{array}\right],\left[\begin{array}{c}{4} \\ {-6} \\ {8}\end{array}\right]\right\}$

Linear Algebra and Its Applications

Catherine R.

Heather Z.

Kayleah T.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Heather Z.

Kayleah T.

Samuel H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications