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1. Let A =and b5Does b belong to the column space of A? Can YOu solve Ax b?it does not belong to the column space because they are not linearly independent(2) What ...

Question

1. Let A =and b5Does b belong to the column space of A? Can YOu solve Ax b?it does not belong to the column space because they are not linearly independent(2) What do YOu expect the projection of b onto W = Col(A) to be?[0,0,0]

1. Let A = and b 5 Does b belong to the column space of A? Can YOu solve Ax b? it does not belong to the column space because they are not linearly independent (2) What do YOu expect the projection of b onto W = Col(A) to be? [0,0,0]



Answers

Show that if the columns of $B$ are linearly dependent, then so are the columns of $A B .$

Here we have given let let the of A. That is this is a linear transformation T. With respect to metrics. City of A. From our square to ask where be a multiplication by A. And lead. Human is equal to 12 and you two is equal to minus one. But we have a two part. So we're here for two different matters. Is of us. In first part we are metrics Is equal to 1 -102. And in part B we have Metrics is equal to 1 -1 -2. We need to determine the said D. A. Of you won T. A. Of youtube is linearly independent in our square. So the solution is given us solution. So let us first find the solution for part A metrics. So in part we have T. is from our two to R. Two. Do you find us D. A. Of X is equal to aim to X? Where metrics A. Is is equal to 1 -102. We need to determine the image of vector U. One and you do so D. E. A. Of human is equal dramatics A. That is metrics it is 1 -1 0 to into. Um so you one is a column vector one to here. We can drive the coordinate as a column vector. So here coordinate you one which is going to cannes Britain as a column vector. Want to after multiplication we get This is required to 14 and D. A. Of Youtube. Which is a called a matrix a. That is 1 -102. Into our metrics be so into our metrics column doctor of Utah which is -11. So here it will be minus two. Um So we get A P. A. So we get D. A. Of human is equal to no we can write we are writing it as a coordinate director. So here it will be -14 and a of you too Is equal to -2. We need to determine whether this said ta of you and auntie of you two is linearly independent in our square. So let the one and care to irony schedulers then given times off The way of you one plus, okay, two times of ta of youtube is equal to zero vector region plie Even times of -14 Plus. Give two times of -2 is equal to 00. Which users minus K one minus two. K two is equal to zero. And the second equation is four times of Caven. That's two times of K two is equal to zero. Now we need to find the value of K one and K two. So we have by initial letter. Uh Very good. So minus keven minus two. K two is equal to zero four. K one plus two. Kato is equal to zero. So minus to get a place to get will be hit cancel. So it will be three Given that zero is equal to zero reach employ K one is equal to zero. No student. The value of Cuban, you know in any of these two week vision regard kyoto is equal to zero. Does given is equal to kato is equal to zero. Which shows that which shows that the way of um And the day of youtube for are linearly independent set D nearly independent sick in art. In our square, no proceeding towards part B. So in part BV higher Mhm. Like T A. Is a transformation from our square to our square defined eyes. The off X is equal to E X were a. Is equal to Our metrics is 1 -1. Yeah minus toto. Mhm. So let us compute the image of you and and Youtube under these metrics. So D. A. Of human is equal to metrics into you and which is equal to What matters, says when 1 -1 minus two to into you and humans it wasn't too. So your tool gives us column doctor So 1 -2 is -1 minus two -2. Plus four is to and P. A. Of you too which is equal to a into Youtube is equal to a Which is 1 -1 -2. Two into you. To our Youtube victor is minus 11 So it gives us 1 -1 to -2. Here it is minus going to minus one is less. Two plus two is four. So you get the two of you will is equal to the magics minus 12 And do you have you to is the magics column matrix -2 and four. So these metrics can be rewritten in coordinate form as the coordinate form you have. Uh huh. The of human Is equal to -12 and D. A. of you two is equal to -2 and four. Oh we check whether it is linearly independent or not. So late. Given here too. Are any skill er that given times of the day of yuan Plus Care, two times of the day of YouTube is a cord zero vector Which imply K one into -1 two. That's catering to minus to four Which is equal to 00 which imply minus keven minus two kato And two K 1. Let's forget to. Is he called 200? I'm comparing we get- Kevin has to care. Two is equal to zero and her it is took it day one Last four K 2 is equal to zero. No, here we see that the second division is two K one plus four K 2020 Which is nothing but two times of Cuban less. Yeah, to kato is equal to zero. Fish imply -2 times of minus K one minus Tokyo two is equal to zero region blow Kevin is Ricardo -2 times of Kato. Thus, yeah, the vectors be A. Of U. N. And ta of you too are not linearly independent. Hence hands the set U. Of U. N. And the way of you too does not form linearly independent. Sick. It is our required solution.

And the question is given us the same statement we have just given up linearly independent. So we're going to determine share. That is we can write it as fast one. Mhm. Israel than one BB square than one thesis work. So even if you're right the size, you can just interchange the rows and columns whenever you have won. A easter one, B B square one. Square can do that kind of thing that will not equal to zero because it has given that linearly independent so we'll find the correct option accordingly. So what we can do here apply here are too tense to Are too negative. R. one and r patterns too artery negative or what it's all it's not you get determinant. The first row remain thing. The kingdom we have 1 -1 deal, Then be -8. Then be square negative. Is where when We have 1 -1 field and see negativity and C square negative is not equal to view. So we'll just keep on going behind this. So expanding along C. One. So we have one times B negative Times c. Plus eight. The negatively then negative being a. See negative A times B plus A be negligible. So we can just protect us now. Here be negatively Is one factor. 2nd factor is the negativity. They're not all of you. We need to get sihpol that okay then negative C. Giving up and then I'm sorry factor out here be negative A background here the negative then we get people did a negative B. Negative so we get the negatively yes Not equal to zero. So we can write this ass to be negative C. Giants. The negativity times integrity that is Not equal to zero. Now which option it is options. He says that the negative six negative eight can live in order to zero. So we go with option thing. Thank you

So to show that beat is in the column space of A. We need to show that A. X equals B has a consistent solution. So to do that, we put A and B in an augmented matrix and do Gaussian elimination. So we'll keep our first row as is and we'll take our second row and replace it with the 1st and 2nd row added together, which gives us a zero a 00 into two. And our third row with the 1st and 3rd row added together, which gives us a zero a negative to a positive two and the two. But we see at this point we have the zero equals two, which is a contradiction, implying that our system is inconsistent. So that means that B is not in the column space of A. For a part A. Then for part B. Again, we'll put our A and B an augmented matrix and then we perform Gaussian elimination to solve this system. So again, we're going to keep our first row the same, and in fact we're going to keep our second row the same as well as it is already zeroed out on the first column. And then we'll take the first row and subtract it from the third row to give us a zero, zero or one, two and 1. And then we'll keep our last row the same as well. Let's go ahead and copy that over to the left and continue. And then at this point we'll cancel a zero the second column. So we'll replace the first column with the first column minus two of the second column, which will give us a one A 0: -4. And uh -1. And then um -2, we'll keep the second column the same. The 2nd row the same. Excuse me. Okay, And we'll keep the third or the same as it is already zeroed out. And then we'll do the glass Toral minus the second row, which is going to give us a zero hero, 01 and four. Now again, we'll copy this to the left to continue our Gaussian elimination and then we'll use the one and the third row, 20 of the third column. Did you do that? We'll do the first row plus four of the third row. Which is going to give us a one, a zero, zero and a seven and a two. Then our next row, we'll do the second row minus two of the third row, Which will give us a zero on one, zero and negative three And a positive one. Our third role will stay the same as well as our 4th row again. We'll copy this over to do the next step of our elimination and then we'll use the last row 20 out the fourth column. So we'll take the first row times negative seven of the last row, which is going to miss a 10 0 -26. We'll do the second row plus three of the last row, which can give us a one 00 13. And then we'll do the third row minus two of the last row, which will give us a 00 a 10 the negative seven and our last row will stay the same. And we see at this point we were able to get to reduce row echelon form, our system is consistent. And so this means that B is in the column space of A. And then we use this last matrix to write B. In terms of the column today. specifically we have that B is equal to negative 26 times the first column of a plus 13 times the second column of a minus seven times the third column of a, then plus four times the fourth column In here. We're getting the 26 from right there, We're getting the 13 from here, we're getting the negative 27 from here, and we're getting the four from there, giving us be in terms of the columns of A.

In this problem, we're given a set of bases and we know the vectors that formed this place is so you know to be and where has to explain why actors are, uh this spaces are the collectors off the on by an identity make. So if this bees are the basis in vector space, be damn You know, that director can be written as a linear combination off these basis vectors. And this it's true before device inspector itself. So we can like be one as one time one plus zero b two plus dr 0.0 b m. So don't be too would be zero times be one plus one time to be too Plus zero times bien I'm gonna be an would be zero b one plus zero view too So old zeros up to be end up Enter Amanda and the answer would be so from this, we can find the coordinated vector for anyone to be one old zeros. And from the second equation, be fine reporter factor for me to to be 010 hold zeros and from 30 question we find the court inspector for intern and director to be 000 old zeros. Why intern? So last term is one says you get sick. Or did that reporter? First time is the first column off the identity matrix for a second when it is the second Poland A or B M for it is defense co


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