if-the-null-space-of-an-8-times-5-matrix-a-is-2-dimensional-what-is-the-dimension-of-the-row-space-o

Question

If the null space of an $8 	imes 5$ matrix $A$ is 2 -dimensional, what is the dimension of the row space of $A ?$

Subham Jyoti Mishra · Accepted Answer

So we have a metrics there. It&#x27;s just 855 metrics. We have been given their Damien son off now space off air. They could have to, and we have to calculate in the dimension off road space off. So the time you know those space off here is equal. Do the diamonds, you know, column space off. If it&#x27;s just a word and minus, which is equally Adminis diamonds and no, no space off. This is coming from the ranked Europe, where Vang off plus diamonds and a lot of yeah, this in the form of payment to end so on his reported a five year You think this the demons, you know your space is getting by minus two, which is a tree.

Runpeng Li · Answer

in this problem, we have Ah, matrix. That is five by six. Bye bye. 68 drinks. And we know that dimension off the nose. Face up. These matrix is five. Sorry for. And we need to figure out the dimension of the rose face A game. We apply the rank, the room first. So the rank off, eh? ISS and minus invention off. Announce yourself, eh? So that&#x27;s six minus four, which is, too, and that it is equal to do you mention up the rose face of a so

Subham Jyoti Mishra · Answer

the question. We have been given a methods here, which is by by six metrics. And we have been given that the diamonds, you know, space a quarter. For now, we have been asked to find off the diamonds and off problem space here. This is equal to rank. Okay. By rank Children, we have, uh and is it Jordan diamonds and all? Nothing. Hey, let&#x27;s rank up here. So therefore there any number off problems with his in the form of him? Cross him. So anything over six years 63 44 less the rain off here which employs the bank off here. It&#x27;s important to therefore the dimension off column space K is toe this two dimensional.

Runpeng Li · Answer

All right, so in this problem, you have matrix. That is seven by six. And we know no space estimation. That&#x27;s dimension, huh? Fire. So we need to find out the dimension of column space. Okay, so the dimension of the columns face, remember our note or in problem? Six. That is a dimension of the column space. Hey, it&#x27;s same as the rank obey. And at the same time, the rank by the lengthy REM rank up, eh? Is and minus dimension up the north face. Okay, so in this case, six dimension of space five. So the column space should have mentioned one.

Subham Jyoti Mishra · Answer

this goes in, we have been given that a six by eight mattress. Now the in order to find Dr Diamonds and off non space. Yeah, we&#x27;ll use around here. Um, so it should be equal in minus, uh, Frank? Well, yeah. This is your man Menace. German Central column space stated now in orderto have this minimum. This will do. This valley is constant is very sorry. Maximum no end is in the form of him cause and so and number of problems. This is eight. So minimum the national. Mm hmm. Is it? 89 s? Make something damaging about the neighborhood. The six? Yeah. Change number off Rose is six. So the next thing that blew off by what columns we do not like civil six. There&#x27;s rank Mexico. There&#x27;ll be six advances.

E R · Answer

were asked to bind the null space of a The definition of the north space is given at the top here with no space oven, Mbaye and meters A is equal to the center vectors X and RN such that the matrix, a times a vector X is equal to zero vector. The problem gives the Matrix A is a two by four matrix shown here and from the A matrix We can see that the value of N is you four which means that we&#x27;re in are for so our except you There&#x27;s gonna be a four by one matrix which looks like next one thanks to next three export, we can start solving this linear equation by first putting a reduced row echelon form by using reduction. So I&#x27;ve re written the matrix A down here in augmented for and starting our reduction. The first thing we do is we can add minus five times read one to road to the breathing abbreviating that we have add minus five both played by row one and adding that through to that is the matrix the first row still the same 1234 in the second row will be zero. That&#x27;s minus five times one plus by zero minus five times two plus six was minus four minus five times three plus seven is minus eight and minus five times four plus eight is minus 12. We still have the zeros on the right hand side. Now we can see that we can get a one of the second column in the second row by multiplying the second row by minus 14 So we multiply. Go to by minus 14 That will give the first row being the same. 1234 Second row is minus 1/4 time, zero minus zero minus 1/4 times. Negative four is one minus 1/4 times minus eight is to and minus 12 times minus one for is three and minus 1/4 times zero is still zero, so we have zeros on the right hand side. Now the next thing we can do is we can multiply. We can get zero. You get a zero here by multiplying the second room by minus two and adding that to that first row and I&#x27;ll abbreviate. That is, we&#x27;re adding minus two times, wrote to minus two tundra to someone. So what&#x27;s blowing that and adding it. We get first rule as minus two times zero plus one, which is still one minus two times one plus 20 when it&#x27;s two times two. Clustering is minus one minus two times three plus minus two or plus four. I mean, is minus two. That was a minus one. There we have a minus two. Their second rules stay the scene. Then we still have the zeros on the right hand side. Now this is reduced to where we can multiply that by our expect. Er in Seoul. So I&#x27;ll rewrite that. Produced a matrix again had 10 minus one and minus two in zero one, 23 Well, multiply that by our four by one. Expect you x one next to next three. The next four. This is gonna equal toe the zero factor which were multiplying a two by four matrix by four by one, which will be a two by one matrix zeros The multiplying this through yet X one minus x three. That&#x27;s zero times x two, which is just zero. So we can leave that out. That&#x27;s an X three that a minus two times X four, which is minus to export, is equal to zero The next one we have a zero times x one zero one times X two two times x three and three times x four is equal to zero. So we have two equations and four unknowns. Then we can solve four x one and x two. Since those are our pivot variables, which have the ones that are matrix the solving for X one, we have that X one will be equal to x three. That&#x27;s to export. We have our to free variables x three and x four and sold in for X two. That will give us money is two x three minus three x four. So with these free variables x three and export, we can said those equal to just some scaler so we&#x27;ll go ahead and well, that X three equal are next four equaled ends. Thanks for people is it&#x27;s now weaken right out. Our expect you for by one. So we have our X one, which we have his x three plus two x four and we&#x27;ll let x three equal are so that old that will be our plus two times X four and exports. Now he would s so that&#x27;s two s. Then we have or X two, which is equal to a minus two x three and x three is arson minus two or minus two are. And then a minus three times x four, which is s minus three s. Let me have where x three, which is our in our export missions s. And so now we can write this out as two factors. But if we factor out this orange s, we&#x27;ll go ahead and backed out. The are and that will give in this first robe that one the second rule we have minus two times over so that we&#x27;ll be minus two minus two in the third row. We have one are that will be one in the fourth row. We just haven&#x27;t s so that will be zero. No, thank you. Out our s and that Who have s in the first rule? We have a close to us. Settle Me too. Second railway have minus three s minus three third rose Justin or so that will be zero in the fourth Rose and asked for me in one Now are vectors. Internal space. Take this form and we can go ahead and write out the answer. And our answer will be that the null space abbreviated by capital in and the null space of a is able to set of actors. Just think this form that we just right out. That&#x27;s sanitation There. Where were our? We have our times one minus two 10 plus s times two minus three 01 such that these scaler Zarin s are real numbers. That&#x27;s the answer.

Problem

If the null space of a $5 \times 6$ matrix $A$ is…

Problem 11 Easy Difficulty

If the null space of an $8 \times 5$ matrix $A$ is 2 -dimensional, what is the dimension of the row space of $A ?$

Answer

3

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Calculus 3 Educators

Lily A.

Anna Marie V.

Kayleah T.

Samuel H.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

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

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Samuel H.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

3

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Samuel H.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Kayleah T.

Samuel H.

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

Linear Algebra and Its Applications