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ProblerDo the following for tne matrices in Exercise:a) Find a basis for the nullspace of the matrix.b) Find a basis for the rOW space of the matrix.c) Find a basis...

Question

ProblerDo the following for tne matrices in Exercise:a) Find a basis for the nullspace of the matrix.b) Find a basis for the rOW space of the matrix.c) Find a basis for the column space of the matrix.d) Determine the rank of the matrix. (Parts (a)-(c) do not have unique answers:)Step-by-step solutionThereris no solution [email protected] this problem Gethele trem @ Chegg subiect exp sk n Aipert

Probler Do the following for tne matrices in Exercise: a) Find a basis for the nullspace of the matrix. b) Find a basis for the rOW space of the matrix. c) Find a basis for the column space of the matrix. d) Determine the rank of the matrix. (Parts (a)-(c) do not have unique answers:) Step-by-step solution Thereris no solution [email protected] this problem Gethele trem @ Chegg subiect exp sk n Aipert



Answers

Determine a basis for the subspace of $\mathbb{R}^{n}$ spanned by the given set of vectors by (a) using the concept of the row space of a matrix, and (b) using the concept of the column space of a matrix. $$\{(1,4,1,3),(2,8,3,5),(1,4,0,4),(2,8,2,6)\}$$

In this video, we're going to find a basis for the span of the following set of vectors. Now, whenever I've been finding a basis for a span of vectors, I always like to take a first initial glance and see if there's any obvious vectors in the set of vectors that air linear combination That is a linear combination of the other vectors in this set to see if there's any obvious linear combination relationships here. Linear dependency, relationships so positivity. And see if you can see any obvious linear dependency relationships in the set of vectors from assuming you've taken a look at it. So the one that stood out to me was that this factor right here, inspector, right here was two times this vector right here. So check this out. The span of this set of vectors the span of the set of vectors is the same as the span off the set of vectors. With this vector removed because this vector is a linear combination off the other vectors in the set, its namely, it's the linear combination of two Times Inspector plus zero times this factor plus zero times this specter so we can remove from the spanning set and get a new set of actors who span is the same as the original is who span is the same as the span of the regional set of actors. So this makes our computations a little bit easier. So we now have easier spanning set to deal with, which are right now. So now we only have three vectors to deal with. We need to find a basis for the span of this set of factors right here. And a nice way to do that as we've talked about in previous bit videos, is to make these vectors the rows of the Matrix. So I'll show you what that means. So this is what I meant. And now check this out. If we find a basis for the road space of this matrix, that's the same. It's finding a basis for the span of the row vectors of this matrix. But the road vectors of this made between made this matrix to have row vectors. That is the same vectors that they're in this set right here. So if we find a basis for the road space of this matrix, that's the same things finding a basis for this. The span of the centre vectors right here, the span of the set effective. So remember, if you want to find a road space based on our previous videos, if you don't remeber, that's totally fine. But let's just review if you want to find a basis for a row space of a matrix. We want to find a basis for the roast base of a matrix. We first we'll reduce matrix to a Roche lawn, form to a row, Rashwan form, and then we take the non zero rose. Take non zero rose. And those non zeros form a basis for the road space off this matrix. Now, now, if you want to know why this method works the details, that's the widest networks. I talk a little bit about how this works in previous videos on previous problems, so check this out. If you have any questions now, positive video and take take a few tries at Rove reducing this matrix. So I'm assuming you've had a go at it. So I'm gonna show the series of Roe Equivalency is that leads us to a row echelon form of this matrix. So this is the series of Roe Equivalency is that leaves us toe this low echelon form matrix and check this out. These are our non zero growth in these two non zero growth. These two non zero low vectors of this row echelon matrix form a basis for the road space of this matrix, which happens to also be based on how we construct this matrix. Ah, basis for the span of this set of vectors which is the same as the spaces for this for the span of this inter vectors. So here is a basis, this set of vectors, the vectors containing the set of actors containing the vector one comma for comma, one comma, three and zero comma, zero comma, one comma, negative one. This set of vectors is a basis for the span of this center vectors right here. So we could do the same thing using the columns using a column space method instead, where we make this set of vectors, we make each of these vectors in the set. The columns in a matrix will show you what I mean by that. So this is what I meant by that. But the column vectors and this majors correspond exactly to the vectors in this set right here. So all we have to do is if we find a basis with a column space of this matrix that's the same is finding a basis for the span off the column vectors in this matrix and the span of the column vectors of this matrix is the same as the span of these, the set of actors right here. So we need to find a basis for the column space of this matrix. But we know how to do that, or it's okay if you don't all review it right here. But check out my previous videos if you want a little bit more of a review as to why this method works, and I also have I will have a lecture video discussing the details of how this meant of how this method of finding a basis of confidence of the Matrix works. But for now, so we first row reduce produce matrix to row echelon for him to locate in the goal of this road reduction series of road reductions to Rochon for Mr Locate, the Pivot columns pivot columns off our original matrix. I'll call this matrix like as I usually do. P Looking to locate pivot columns of P in the second step is to take those pivot com's take Pivot Collins and those pivot Collins take the pivot calms as basis vectors. Take Piva columns in those Piva Collins of P that we located by Rhodesian P to a row echelon form. There are all call that revolution former produce matrix to ocean form P prime P prime. So the Pitta columns of P will form a basis for the column space of P. So the pivot Collins of P will form a basis for the con space of P. So take a minute, positivity on road use this matrix to a row, Ashlan for So I'm assuming you've had a go at it. So all reveal now the Siris of Roe equivalent sees that leads us to a row echelon form of this matrix. So this right here is the series of race series of Roe equivalent sees that leads us to a rational in form of the matrix penal. Call this relation on foreign p prime and this reveals where the pivot columns of PR located the difficulties of P. exactly correspond to the pivot calms api prime. So in this case, since the first and second columns of P Prime are the pivot columns or are the pivot columns of P. Prime because those are the columns that contain the leading entries, then that means that the first and second columns of P are the pecans of Pete. Through these two column vectors form a basis for the column space of the Matrix P. And so we have our basis. Or we have a basis for the conference API, which is the same, which is the same by by how we constructed the Matrix p as, ah, basis for the span of the set of actors in that and again, that's the same as a basis. The span of this set of actors since the span of these two sets of actors are the same. So the set of excuse one comma contain the CEP containing the two vectors, one comma, four kind of one, comma three and to common eight comma, three comma five. This set of actors right here is a basis for the common base of P, which is the same as span of this set of vectors, which is the same as a span of this set of vector

If we want to find a basis for the span of the following set of vectors than one way to do this is to make these vectors the row vectors of a matrix and then find a basis for the row space of that matrix. Another way to do this is to make thes vectors the columns of a matrix and then to find a basis for the calm space off that matrix. The reason why these two methods work is because, say, let's take the road space method method first. If you want to find a basis for the road space of this matrix, we would be That's the same Assane based off the definition of a row space which exactly which the definition of a road spaces the span of the row vectors of matrix. So finding a basis, the road space of this matrix right here we would be finding the basis for the span of the road factors. And the row vectors are exactly the vectors in the set. And same for the calm space. If we wanted to find ah basis for the column space of this matrix and what this is green, Let me change that real quick. So if you want to find a basis for the column space of this matrix, we would be finding a basis for the span of the column Factors, because the definition of a con space of a matrix of the span of the contractors and the contractors of this matrix are exactly the vectors in this set. So let's use the roast based method to find a basis for the span of the centre vectors right here. And the roast meats method involves first row, reducing the matrix to a row echelon form of the matrix of reproduce matrix to row echelon form to a row echelon form. And then second, we take the non zero rose to take the non zero rose of the Rochon for matrix. And those non zeros were formed a basis for the road space of this matrix. And for an explanation as to why this works, you can check out the previous videos on finding rose spaces off right, rather than a basis for the road space of the Matrix. The positive video and rove reduce this matrix right here to a row Ashkelon form of this matrix. I would let you do that composite video. So I'm assuming you've had a go at it. So first, all reveal the roe equivalence ease that leads us to a row echelon form of this matrix. So here is the series of Roe equivalency. So that leaves us toe a row echelon form of this matrix right here. So this is our low echelon form and all we have to do is take the non viewer rows of this Rolla Schwann farm for this row in this row and these two rows. The set containing these two roads is a basis for the span of the roast base of this matrix. And the roads base of this matrix is the same as a span of this set right here. So we found ah, basis for the span of the set. By here, that basis is the set of vectors containing the one vector or the two vectors. One of them is one comma, three comma three and the other one zero comma to common. Negative four. No, to use the con space method, we do the same thing. We find a basis for the calm space of this matrix and then we use. That basis has also because the basis for the common space of this matrix is also a basis for the span of this set of echoes right here. Because the common space of this matrix is the same as the span of the set of vectors. So to find a basis for the confidence of this matrix, we first produce produce matrix to a row echelon form, echelon form to find to find because our goal when we use the common space matters, we want to find the pivot Columns of this matrix will call this Matrix P. I want to find the pivot calms of P to find pivot columns. I love P in the second step is to take those pivot calms in those Pippa calms the pivot columns of the original Matrix P take the pivot column in the pit Occam's of the original the this matrix right here the pecans of this matrix were formed. Those column vectors will form a basis for the calm space off this matrix to take the pivot column. So posit video and take a second and row produce this matrix. So I'm assuming you've had a go at it I'm gonna reveal Reveal the set of or the the Siris of Roe equivalency is that leads us to a row echelon form of this matrix. Now, these are the set of or the series of low equivalent sees that leads us to a row echelon form of the matrix P of this matrix right here. And this is in row echelon form because all of the zero rose there at the very bottom of the Matrix. Each leading entry is to the right off the leading entry in the row above it. And each leading entry has zeros directly below it in the Colin of that leading entry lives in. So we know now because this matrix is in Rochel isn't a Rochon form. We know where the Pitta columns of this matrix art This column In this calm the columns with the leading injuries. The columns with the leading entries in the pivot columns in this matrix right here call this Matrix p Prime. The pivot columns and P prime are exactly the same as the pivot cons and P. So we know that the first column and the second column of P. R. The Pitta cons of p. And based on our based on our method, that means that they're the vectors. The column vectors that form a basis for the cons. Basic p. So the vectors one comma, three comma, three and one comma, five comment native one are the vectors. Or rather, this is the set of factors. This is a set of vectors that is a basis for the column. Face off P in the common space. API is the same as the span of the set of actors right here, now and again for a review as to why this method works I have posted on. I will post a video in my lecture video series explaining a little bit more why this method works a little bit more complicated than tha scope of this video allows. So, yeah, so this is a basis for the confidence of P in the column. Space of P is the same as the span of the set of actors right here

If we want to find a basis for the span of the following set of vectors than one way to do this is to make these vectors the row vectors of a matrix and then find a basis for the row space of that matrix. Another way to do this is to make thes vectors the columns of a matrix and then to find a basis for the calm space off that matrix. The reason why these two methods work is because, say, let's take the road space method method first. If you want to find a basis for the road space of this matrix, we would be That's the same Assane based off the definition of a row space which exactly which the definition of a road spaces the span of the row vectors of matrix. So finding a basis, the road space of this matrix right here we would be finding the basis for the span of the road factors. And the row vectors are exactly the vectors in the set. And same for the calm space. If we wanted to find ah basis for the column space of this matrix and what this is green, Let me change that real quick. So if you want to find a basis for the column space of this matrix, we would be finding a basis for the span of the column Factors, because the definition of a con space of a matrix of the span of the contractors and the contractors of this matrix are exactly the vectors in this set. So let's use the roast based method to find a basis for the span of the centre vectors right here. And the roast meats method involves first row, reducing the matrix to a row echelon form of the matrix of reproduce matrix to row echelon form to a row echelon form. And then second, we take the non zero rose to take the non zero rose of the Rochon for matrix. And those non zeros were formed a basis for the road space of this matrix. And for an explanation as to why this works, you can check out the previous videos on finding rose spaces off right, rather than a basis for the road space of the Matrix. The positive video and rove reduce this matrix right here to a row Ashkelon form of this matrix. I would let you do that composite video. So I'm assuming you've had a go at it. So first, all reveal the roe equivalence ease that leads us to a row echelon form of this matrix. So here is the series of Roe equivalency. So that leaves us toe a row echelon form of this matrix right here. So this is our low echelon form and all we have to do is take the non viewer rows of this Rolla Schwann farm for this row in this row and these two rows. The set containing these two roads is a basis for the span of the roast base of this matrix. And the roads base of this matrix is the same as a span of this set right here. So we found ah, basis for the span of the set. By here, that basis is the set of vectors containing the one vector or the two vectors. One of them is one comma, three comma three and the other one zero comma to common. Negative four. No, to use the con space method, we do the same thing. We find a basis for the calm space of this matrix and then we use. That basis has also because the basis for the common space of this matrix is also a basis for the span of this set of echoes right here. Because the common space of this matrix is the same as the span of the set of vectors. So to find a basis for the confidence of this matrix, we first produce produce matrix to a row echelon form, echelon form to find to find because our goal when we use the common space matters, we want to find the pivot Columns of this matrix will call this Matrix P. I want to find the pivot calms of P to find pivot columns. I love P in the second step is to take those pivot calms in those Pippa calms the pivot columns of the original Matrix P take the pivot column in the pit Occam's of the original the this matrix right here the pecans of this matrix were formed. Those column vectors will form a basis for the calm space off this matrix to take the pivot column. So posit video and take a second and row produce this matrix. So I'm assuming you've had a go at it I'm gonna reveal Reveal the set of or the the Siris of Roe equivalency is that leads us to a row echelon form of this matrix. Now, these are the set of or the series of low equivalent sees that leads us to a row echelon form of the matrix P of this matrix right here. And this is in row echelon form because all of the zero rose there at the very bottom of the Matrix. Each leading entry is to the right off the leading entry in the row above it. And each leading entry has zeros directly below it in the Colin of that leading entry lives in. So we know now because this matrix is in Rochel isn't a Rochon form. We know where the Pitta columns of this matrix art This column In this calm the columns with the leading injuries. The columns with the leading entries in the pivot columns in this matrix right here call this Matrix p Prime. The pivot columns and P prime are exactly the same as the pivot cons and P. So we know that the first column and the second column of P. R. The Pitta cons of p. And based on our based on our method, that means that they're the vectors. The column vectors that form a basis for the cons. Basic p. So the vectors one comma, three comma, three and one comma, five comment native one are the vectors. Or rather, this is the set of factors. This is a set of vectors that is a basis for the column. Face off P in the common space. API is the same as the span of the set of actors right here, now and again for a review as to why this method works I have posted on. I will post a video in my lecture video series explaining a little bit more why this method works a little bit more complicated than tha scope of this video allows. So, yeah, so this is a basis for the confidence of P in the column. Space of P is the same as the span of the set of actors right here

If we were given the following set of vectors and if you were asked to find a basis for the span of this set of actors in one way, we could find a basis for the span of the set of vectors is to make this vector the first row of the Matrix, this vector, the second rover matrix in this vector, the third row of the major. That's to say, make the rows of the matrix correspond to each of these vectors. So let me show you what I mean by that. So what we did here is we made a matrix whose rose corresponds exactly to the vectors in this set right here. So check this out. Based off of the definition off aerospace, off a matrix, The roads base of the Matrix is the span of its row vectors. So since the row vectors in this matrix correspond exactly to the vectors in the set of vectors, if we find the basis for the roast base of his matrix, we will have also found ah, basis for the span of the set of victories. Because again, the row space is a span of its row vectors. So finding a basis for the aerospace that this matrix is the same spine. Ah, basis for the span of the set of vectors. So to find a basis for the most base of this matrix, we first we first row, reduce ro reduce matrix to a row echelon for and then next we take the non zeros take the non zero both in those lows. Those non zeroes off the row echelon form matrix form a basis for the road space of this matrix. And the reason why this works, I discussed in previous videos where we talk about finding the basis for the road space of the Matrix. So if you're curious about why that works, go ahead and check those out. So take a minute. Impossibly dio and load is this metric toe a row, ash on form. I'm assuming you've had a go at it. I'm gonna reveal the Siris of low equivalent sees that leads us to a Rochon form of this matrix. So these low equivalence ease leave this to or this Roque, the Siris of Roe equivalent sees, leads us to this row echelon for matrix right here in the non zero rows of this role echelon for matrix. This room right here and this one over here Thies to row vectors form a basis for the road space of this matrix, which is the roadways of this matrix is the same as a span off this set of vectors. So a basis for this set of the span of the set of actors is the set of exes containing the vector one comma, one common negative, one comma too. The first non zero grow in this revolution for matrix and a vector zero comma Negative one comma five common negative eight. No, we could do the same thing, but using the notion of a column space. So if we made this vector rather, these vectors in this set correspond exactly to the column vectors and in Matrix. So let me show you what that means. So the column vectors in this matrix correspond Exactly. You are exactly the vectors in this set of vectors. So that means if we find ah basis for the column space of this matrix, the Mueller also found a basis for the span off the set of vectors because the column space of this matrix is the span of the car vectors and the column vectors in this matrix are the same as the vectors in this set of vectors. So take a minute. Posit Video Road Is this major or rather? Let me show you that. Let me first review the method off finding a basis for the class based on a matrix. So we first wrote, Reduce matrix to a row echelon form a row echelon form. We'll call that Rochelle on full on P Prime and let's call this Matrix P, as they usually do so, produce major toe Aurora Sean form P Prime to find the pivot columns to find the pivot columns of P. Because the pivot columns of P Prime correspond exactly to the pivot columns of P, Let's say I found out that the pivot cons of P Prime are the first and third columns of P prime. Well, then, that means that the first and third columns of P are the pivot columns of P. So once we find those pivot cons, we take the pivot columns of P of the original Matrix B. This matrix right here, this matrix right here we take the pivot cons of peace or take pivot columns of P and those columns form a basis for the calm space of P. So all lets you find a row echelon form of this matrix p to go ahead and possibly dio. So I'm assuming you've had a go at it. So I'm going to reveal the Siris of Roe. Equivalency is that leads us to a row echelon form of this matrix p. So this is the series of Roe equivalent sees that leads us to a row echelon form of the matrix P of this matrix pee right here. And this is our role echelon form. And this role echelon form reveals where the pivot cons appear located. Because this column in this calm or a pivot com off this relation gonna call this relish on for matrix p prime for this column in this column Occam's with leading entries comes one and comes to are the pivot comes a p prime. So therefore, the first and second columns of P. R. The pivot calms API, and we know that the pivot columns of P former basis for the calm space of P and the common base of P is the same as the span of the set of vectors. So we found a basis for the span of the set of X right here, namely the set of vectors containing the vector one comma, one common negative, one comma too. And the vector to come One comma, three common negative four in the set of vectors right here is a basis for the calm face of this matrix, which is the same as a span of the set of actors. Now, you might be wondering, Why does this fan of the set of vectors have two different bases? But remember, So first off, first off, this span of the set of vectors is a subspace of our four. Why is it a subspace of our four? Because the span of the set of vectors any vector in the span of the set of vectors is a linear combination off these vectors and Angelina linear combination of these vectors will contain four rial number components. That is, it is a vector in our for the set of all vectors containing four real number components. So remember, a subspace can have multiple bases When he said it again. A subspace can have multiple bases and Here's a good example of a subspace having at least two bases. They can have more. It can have more basics. It doesn't have to have one basis. So here are two bases for one vector, one subspace of our four, namely the span of the center vectors right here.


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