in-exercises-3-6-find-an-explicit-description-of-nul-a-by-listing-vectors-that-span-the-null-space-4

Question

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{rrrrr}{1} & {5} & {-4} & {-3} & {1} \ {0} & {1} & {-2} & {1} & {0} \ {0} & {0} & {0} & {0} & {0}\end{array}ight]
$$

Doruk Isik · Accepted Answer

his problem with us to find the bacteria. That&#x27;s fun. The no space. So we&#x27;re giving this a Metrix, and I&#x27;ll start the solution by being a single rule operation. Simon Sire Number one will be the origin. A little row one last five times Road to And that would be me and you Amy objects. And that will be 10 since native age. One 01 made it 210 and 00000 So we have five columns and imagine we have five variables because we&#x27;re looking for a solution off for eggs is zero unless assume x factor to be explore next two eggs. Three. Explore an ex So let&#x27;s go work this man juice. Um, using this extractor in a subtle being near you creations. So the pressure will give us explore plus six x three, minus eight x for plus expired zero and then the second room will give us X two when it&#x27;s two x three. Plus explore is equal to zero so we can ride our music stuff X as x one x two x three Export expire and that will be, um, extreme NATO six two 100 plus it&#x27;s for eight Negative. 1010 plus, like five, Um, negative 10 zero 00 So then director suspended Mills space will be negative. 6 to 100 Negative eight Made 21010 and Native 1000 zero.

Oumeng Zhang · Answer

all right when we first looked at, It&#x27;s crashing. We need to think about what&#x27;s the definition of a null space. So firstly ah, are you a point here? Definition of what No space is all the combination Oh, the&#x27;s X which tracks that all ex pants matrix A We&#x27;re equals zero on all the combination of these kind of wax. It&#x27;s a no space of matrix C and the next question we need to think about it is waas the dimension of no space. Uh, here we knew. Some predict the knowledge that the rank of this matrix A equals 23 on because a three by five metrics sold the dimensional No space were ICO 25 minus three, which equals to two. And that means no space of A it&#x27;s ah, two dimensional space. The vector we are looking for that we need to use two independent basis or two invented factors to represent this dimension. So falling the definition we just need to fund this matrix so called a B c d e the damn metrics times this it would say Well equals two help. Ah, Victor, contents read zero. So we simply solid this I in your system. So that&#x27;s it. Christian will help on. We have note, uh, you were always equal to their. So to solve a second equation we need to find Ah, see, my i d. Coastal zero. If we assume that C equals zero, so be will be one and then having joke Adam for a secretion here and we assume that B equals two there. And then we&#x27;ll have this a ico to minus four. And this solution will be one business. Oh! Oh! One description off on space of me. Trixie. So put it down here. He equals two minutes. Four be equal. Stare and sing. Wait, That&#x27;s the first answer you have. And still we&#x27;ll be called Dimension of no space Away it too. So we are looking for another solution. Still, if we assume that&#x27;s the Eagles to lie, uh, so the will be the same. But this time, if you have another solution for this parameter B on B, let it go to one. In that way, they will be co 22 So if we write down, you have our second solution. And this rap helped his question with two ah description for the no space of Trixie

Doruk Isik · Answer

in this problem here has to find that curious. That&#x27;s the spice over. Basically trying to find Dr Eggs or Eggs is eight times likes is zero. So let&#x27;s court this metrics Aiken to set a few questions so mentioned, we have X and X is explore next to X three innings. Cool. So then a eggs would be for the next set of equations Ex form by a sect six x two plus four extreme is zero and cheers to start to x three. Is it because you and what you have is we have extort? Explore. These are free. Where you so from this be fine extreme to be zero. And if your blood gets 30 right here, we find that explore is take up to six x two so that we can write x as explore next to x 34 and that is equal to X to times six 100 Um, plus x four times 000 So then the vectors that spends pace we&#x27;re being 6100 and 000

Michael Jacobsen · Answer

In this video, we have a two by four matrix as provided in our goal here is to compute the null space of a well first note that the null space of a set is a set that consists of all vectors, such that eight times X is the zero vector. So, in order to find the null space of a we take this matrix equation and go to augmented matrix will be 10 31 54 zero negative too. And we&#x27;re augmenting with the zero vector. So that will go here. We didn&#x27;t have to augment with Israel Vector because we know that this right most column is always going to be zeros. But sometimes it can be handy to include it anyways, so that we recall that will have variables x one x two x three and x four, and that this is not the side that contains the right hand side corresponding here. So let&#x27;s begin Row reduction. We have a pivot in this position was zero below, So we&#x27;re all set there. In this pivot, we just need to eliminate the entry above, which is three. So let me start by copying this matrix I&#x27;ll copy Wrote To, which is 014 negative, too augmented with the zero vector and will multiply row to buy negative three. At the results to Row one, we obtain one zero negative seven six and then a zero. At this stage, we&#x27;re now in real reduced echelon form and so we can read off from Row One. That x one minus seven x three plus six x four is zero from Row two. We have that x two plus four x three minus two x four is zero. Then the X three and X four correspond to free variables. So I will just write to that x three equals x three and X four equals x four. Just as a way of saying that there&#x27;s no restrictions for these variables. Now we&#x27;re rate. Just show what the null space A is. First, let&#x27;s say if a X equals zero what the solution X should be. Well, we have that. There&#x27;s two free variables x three and x four. So it&#x27;s going to be x three times a constant vector plus x four times another constant vector. And the way we produce these constant vectors is we imagine solving these equations for X one and x two We would have four x one x one is equal to seven x three minus six x four x two is negative four x three plus two x four and no restrictions on next three. Saul Ray X three is x three and similarly. All right X four is X four again now to determine the first factor we have over here on the solution. We just pulled coefficients for X three, which are seven negative 41 and zero. Likewise, coefficients for X four are negative, six to zero and one. And because there&#x27;s no restrictions on X three and x four, weaken right here, x three x four are in the set of real numbers. We&#x27;re not done yet. Remember, our goal is to compute the null space. So this is what we&#x27;ve shown the null space of this matrix A is equal toothy span, which means it&#x27;s a vector space, and the span consists of two vectors. They&#x27;re the constant vectors. Seven. Negative 410 negative, 6 to 0 and one. So that&#x27;s the null space for this particular matrix.

E R · Answer

there has to find the null space of a definition of the no space is given at the top here with the null space oven n by n Matrix A is a set of vectors x and or in such that the matrix a times a vector X is equal to the zero victory grown gives the matrix A is a three by four matrix Showed you we can see that then is he got a four Which means that we&#x27;re in our for which means that our expect here is gonna be a four by one matrix with X one x two next three and export And now we can solve this linear equation by first producing the Amy Tricks to roll Ashland for and so we&#x27;ll start that reduction. I&#x27;ve re written the matrix A here, and we can start by seeing that road to we can make this term zero If we multiply Rou won by three and and that&#x27;s we&#x27;re going to Serena and minus three times were one to so where one will stay the same. 13 minus 21 zero, multiplying room one by minus three and adding it to we have minus three times one plus three zero most three times three I was 10 is one minus three times minus to plus and minus for is to and minus three times one plus six is three in my eyes, three times zero plus zero is what is zero and our third rose days This thing 25 minus six minutes. 10 Now we can try to make this term zero. We can do that. If we and the first row. If we add we had minus two times the first row to the third road, we&#x27;re gonna add minus two times the first row to the third room, so our first row stays the same. Our second rest is the same. Our third Rosa will mind multiplying minus two comes the first. Roman added through threesome minus two times one. It&#x27;s minus two plus two is zero. Then we have minus two times three plus five is minus one minus two times minus two plus minus six Here is minus two and minus. Two times one plus minus. One is minus three in my eyes to time zero plus 00 See, Now we can we can eliminating this third row if we just add the second row to the Third World. So add one times were 203 We&#x27;ll have to scroll down here. But you can see that adding the second row here to the third rule will just make third row all zeros. We&#x27;ll go ahead and write that out. Our first drove stays the same 13 minutes to 10 Second row stays the same 01 23 hands, adding or second room to our third rule. We get old zeros. It&#x27;s now we can get rid of this three here and make that a zero. Multiply minus three by the second row and and that&#x27;s in the first room. So add minus three times or a two to Rawood, so we&#x27;ll have zero times minus three. Plus one is one minus three times one plus 30 minus three times two plus and minus to the minus eight. The minus three, the minus three times three plus one is minus eight in their second room. It is the same in our third stays the same. It&#x27;s knowing. Now we have it in reduced. That&#x27;s the long form produced Rochel inform. So now we can go ahead and do our matrix multiplication, so rewrite are reduced. A girl notice right out our linear equation. A X equals zero. So we have our A majors produced. It was one zero 10 minus eight minus eight 0123 0000 Homes are extractor x one x two x three Next four and that is equal to R zero Victory, which is a three by one majors. Now multiplying is out. We have her first row one times x one she&#x27;s x one zero times x two and then a minus eight times x three. That&#x27;s a minus, But it&#x27;s a A minus eight next three and the minus eight times X for is equal to zero. Let me have zero times x one. We have a two a one times X to his ex too Two times x three and three times Export missing grow zero in our third room is old zeros. So we have two equations and four loans we can solve for X one Little soldier X one that equals eight times x three plus eight times x four Muscle for X two is equal to minus two x three and minus three x four Now I can let x three and x four There&#x27;s a free variable so we can let those equal just any scaler. Well, uh, x three equal Any scaler are in x four equal any scaler s. It&#x27;s now we can write out our X factor for X one is eight x three plus eight x for which is Pete Orr plus eight s, x two, which is minus two X three months, three explores and monies to our minus three as let me have X threes are next fours? That&#x27;s no. I can write this out as two separate Victor&#x27;s If we just factor out the or in the S, that will be equal to our times. Aids Hard times minus two 10 was s times eight minus three zero and one. So are vectors of make up this form in our no space so we can go ahead and write out our solution which will be the null space. Just capital in a is equal to set of actors set of factors which take this form or a minus 210 What s hey minus 301 Such that or an s are real numbers. This is the answer

Doruk Isik · Answer

in this problem. We&#x27;re asked to use men Take saying from exercise tree and they were asked to find knows your victory and no, they don&#x27;t. They&#x27;re vectoring problem. Hey, so many column is actually a nuncio vector in call eight. So is an answer to properly It could be any Cole. No, partly a time. Thanks. I should say zero. So then the system is your enroll. Echo for will be 1001 negative seven or 6200 The way I got this is I said near Rome or will be row one minus three row too. So now we have a system off recreations here. We can say that Explorer is equal to seven x three Minus execs or and x two, The sequel to Negative four X Tree. It&#x27;s a negative two plus two eggs for now. Thanks. What? The next two depends on x three MX four but X three units for our our free breakables. So plus, assume a three is one and x four is one that from you questions second equation you find next to to be 94 plus do like a dream. And from the 1st 1 We plan X one to be 7.6. That is one. So what? And we find the zonal director to be one naked too. 11

Problem

In Exercises $7-14$ , cither use an appropriate t…

Problem 6 Medium Difficulty

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{rrrrr}{1} & {5} & {-4} & {-3} & {1} \\ {0} & {1} & {-2} & {1} & {0} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right]
$$

Answer

Not a vector space.

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

Linear Algebra and Its Applications

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Not a vector space.

Linear Algebra and Its Applications

Lily A.

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Vectors Intro

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

Lily A.

Heather Z.

Caleb E.

Kristen K.

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

Linear Algebra and Its Applications