m-determine-whether-mathbfw-is-in-the-column-space-of-a-the-null-space-of-a-or-both-where-mathbfwlef

Question

[M] Determine whether $\mathbf{w}$ is in the column space of $A,$ the null space of $A,$ or both, where 
$$
\mathbf{w}=\left[\begin{array}{l}{1} \ {2} \ {1} \ {0}\end{array}ight], \quad A=\left[\begin{array}{rrrr}{-8} & {5} & {-2} & {0} \ {-5} & {2} & {1} & {-2} \ {10} & {-8} & {6} & {-3} \ {3} & {-2} & {1} & {0}\end{array}ight]
$$

Doruk Isik · Accepted Answer

in this problem were given a directive W and then they say, and nowhere else to turn with the new is indeed home space or and whether it is indeed no space, eh? So I used that to do that. I used our recommitment to find the role Ackman form off the augmented system A that you could use meth at this moment. Or you could use any rental to find the oracle for your vent system. And we end up getting this matrix. Uh, 108102 01 negative. 203 000 Or Warm and waas Wrong years old zeros. And as you can see from this, our lamented metrics aerobic purple X because be this system is court system. So this means that w is actually mean home. A got to take 40 check weather, W&#x27;s and no pay or not. We&#x27;re gonna multiply, Amy, don&#x27;t you? And if you do so you end up with a vector off zero. So this proves that the blue is also in no

Viswesh Uppalapati · Answer

in this video only solving problem. Number 36 of section 4.1 Grand word. Uh, we&#x27;re asked to determine if why is in the subspace of our four spend by the columns of a where why is, um, why is already in our for a zit has four entries given here, and but we need to see if it&#x27;s spend by these specific columns in our four. So in a better part of the matrix, a first we need to set up in augmented matrix here, um, to solve this system. So we just combine the two matrices are to, uh, the Matrix and the vector. So here we have 38 negative five to as the first column. Negative five seven. Negative eight. And negative, too, was the second. Call them negative. Nine Negative 63 Negative nine. Is that their call? And then a line in between. Negative four. Negative eight six and negative five. As the last column as this is the B column in the Matrix equation, X equals B in resolving for a set of scale er&#x27;s X, where any where x one multiplied by the first column and x two multiplied by the second call in the next three multiplied. But third call them would equal why so this season are you have calculator and to make your life easier sense Doing this by hand would take a long time. But, um, so you get if you saw this, you should get and put it in our area. We&#x27;re reduced to ash lawn for me. Should get negative one on DDE 3/5 on zero. So, um, we solved for three values your negative 1 52 53 5th where? Negative 1/5 times. What&#x27;s called this? The one or a one 81 83. So the night of 1/5 time&#x27;s a one plus negative to fifth times a two and was 3 50 times a three equals. Why? So this, um since there is a solution to this system, you know that we know that, um, Why is spanned by these columns the columns of this matrix, and are for

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.

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

E R · Answer

were asked to find the null space of a definition of the no space is given at the top here, with the no space of an MBA and matrix is given by the set of vectors eggs and are in such that the matrix, a times the Vector X is equal to the zero vector. The problem gives the Matrix and he has a three by three matrix shown here. We can see from there that value of N is three, which means we&#x27;re in or three so we can write our expect er as a three by one matrix with X one next to the next three. So now we can start the soul of this linear equation, but using reduction to get a in a row echelon form. So every Retin A and augmented form down here we can start our row reduction by seeing that we can get, we can get a zero here If we multiply and minus four times the first row and add that to the second room, we&#x27;re gonna add what is four. I&#x27;m drew one and added to that gives our first rule stay the same. 11 is 21 and uh, zero. So I have a minus four times one, and it&#x27;s a four, which gives zero. We have a minus four times a minus to plus and minus seven, which gives warm. We have minus four times one plus of minus two, which is minus six. Their third road stays the scene minus 134 Then we have a minus four times zero, which is zero in the zero of the third room. So now we can We can get a zero here if we add one. Times were one to row three. So add one times were one the road three. So that will give our first rebels stay the same. Their second rule stay the same on our third row will be one plus and minus one, which is zero. Then we have ah minus two minus to plus three, which is one. And then we have one A C one plus four, which is five. Then a 000 Now we can get let&#x27;s see weaken, multiply the second row by minus one and that the three and that will give us a zero here, several multiple it or we will we will end minus one one&#x27;s road to and and that the three So our first row stays the same or a second room will stay the same. 01 minus 60 Then our third rule have a minus one times zero, which is zero minus one minus one times one Edits the 10 the minus one times six plus five which is 11 the most one time 00 which is zero. Now we can dio one more step. Uh, to get a one year, we can multiply that third row but one ever 11 for most playing the third row by one every living We&#x27;ll show that down here So our first true will stay the same one minus 210 2nd row stays the same and zero minus one minus six zero in our third room will be zero zero and 1 11 times a left. 11 is one and zero. So that&#x27;s our Rochel Inform Matrix A And now we can multiply that by or expect Cherie because x one x two x three that is equal to zero factory which is a three by one matrix was here s so that we can multiply this out that will have first row x one minus two times x two and a one times X three is equal to zero. Next, we have a zero times x one minus one times x two the minus six times x three 00 and then we have zero times x 10 times X two and one times x three is equal to zero so we can see that x three equals zero. If we played that in here than X two has two equal to Z equals zero. Flex two is equal to zero. This was an ex too. So if x two and x 3/2 the equals zero, the next one has to equal zero that are expected has one solution which is, well, zeros. So we can write the solution to our problem that the North space of a because one solution which is zero factory. That&#x27;s the answer

E R · Answer

were asked to find a null space of a and the definition of the null space is given at the top here with the null space oven. N by N Matrix A is equal to the set of vectors X in or in such that the Matrix, a times the Vector X is equal to the zero vector. The problem is matrix A as a three by two matrix, with the first column being to one minus three in the second column being minus forward to my Spi we can see from are a matrix that are in value is equal to two, which means we&#x27;re in or two, Which means our expect er is gonna be a two by one matrix with X one and X two. Now we can go ahead and solve this mystery equation. Um, then we&#x27;ll do that by putting a in Rochel informed by using road reduction. So we&#x27;ll start with the reduction and I have the Matrix a re written down here. So this is the matrix. Every rate rewritten and augmented form of zeros on the right hand side. And to reduce this into a relational, inform the first thing we can do is per mutate Row two with row one So and I can abbreviate that with a P for perm et row one into and that will give the new Matrix 12 through minus forward and minus three minus five zero is still on the right. No, the next thing we can do, we can We can get a zero in the bottom left hand corner. If we add three times row one to that third red making abbreviate that within a for ad, we&#x27;re gonna add three times row one to row three. So that will give us. Everyone will still be seen for a two will still be the same. Now we&#x27;re adding three times were ones that were three times one plus minus three, which is zero. Let me have three times to, which is six plus minus five, which gives us one and our zero&#x27;s air. Still in the right hand side, we have three times zero plus zero is zero Now the next thing we can do is we can get zero on the first row are on the first column of our second row. If we add minus two times, red ones are zero to. So you know, abbreviate that is adding minus two times were one. So to that&#x27;ll give are saving through one. Let me have minus two times one plus two, which is zero. Then we have minus two times two, which is minus four plus minus four, which is minus eight. And our third row stays the same and or zero stay the same. Now this matrix is reduced enough to where we can sold it. I&#x27;ll rewrite it down Here, begin. We&#x27;re solving linear equation. The Matrix. Eight times. The X is equal to 00 vector. And so we have our reduced me reduced matrix A as 120 minus AIDS and 01 And we have our expect. Er with excellent next to that&#x27;s equal to zero. No, that looks like 10. Or rewrite that. Yeah, that&#x27;s equal to director. So multiplying this l we get x one plus two x two equals zero. We have a zero times x will in which zero minus and eight times x two is equal to zero. We have a zero times x one, which is zero and a one times x Two years of X two equals zero we have this x two equals zero. And if we pulling that into our second equation, that still just zero equals zero. It looked like that appears X two equals zero. We get that X one is also you go to zero so x one equals zero. The next two x two equals zero and that is one solution. So are no space contains only contains the zero factory for eggs is just zero bacteria. It&#x27;s a two by one matrix. Double zeros. Now we can go ahead and write our solution out for the middle space for a solution will be that the null space abbreviated by capital in in the no space of a is equal to zero vector. This is the answer.

Problem

$[\mathbf{M}]$ Let $\mathbf{a}_{1}, \ldots, \math…

Problem 38 Hard Difficulty

Answer

$w \in \operatorname { Col } A$

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

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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$w \in \operatorname { Col } A$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications