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LSRLI2. (10 points). Find a QR factorization of matrix4 6 8 2 ~4 ~6 0 15by completing the following stepsUse the Gram-Schmidt orthoganalization algorithm to find an...

Question

LSRLI2. (10 points). Find a QR factorization of matrix4 6 8 2 ~4 ~6 0 15by completing the following stepsUse the Gram-Schmidt orthoganalization algorithm to find an orthogonal basis for the column space ofNormalize vectors obtained in the = previous part . Form matrix Q using vectors obtained in the previous part Express the corresponding matrix R in terms of A and Q Find entries of R

LSR LI 2. (10 points). Find a QR factorization of matrix 4 6 8 2 ~4 ~6 0 15 by completing the following steps Use the Gram-Schmidt orthoganalization algorithm to find an orthogonal basis for the column space of Normalize vectors obtained in the = previous part . Form matrix Q using vectors obtained in the previous part Express the corresponding matrix R in terms of A and Q Find entries of R



Answers

Find a QR factorization of the matrix in Exercise 12

In problem 15. We want toe. Make a Q R factory ization for these metrics. The first system is to apply Graham ish minted process. So get orthogonal basis for the columns in this video. That's a ploy. Gram Schmidt, We get the one as the first victor on minus one minus 111 And we get V two as the projection as the second victor for the second column 214 minus four to and we subtract the projection for the second victor in tow, V one on minus one minus 111 We multiply by a factor to make this projection projection. This factor has in denominator doesn't product between these two victims. You have to minus one minus four. Minus four was two equals minus five. Divided by the product off. We want bond itself, which gives life equals two plus 13 one one plus two one plus one equals two. You have one minus minus minus is one minus one equal 00 one minus one equal zero. We have full minus minus one equals three minus minus minus equals minus. Then we have minus four minus my minus minus. She's plus once four plus one equals minus three. And finally we have two plus one equals three. This is V two. Then we get with three. As the third column in the metrics, which is serve. The column is five minus four, minus three five, minus four minus 371 minus the projection off this victor onto V two, 30333 Want to blow it by a factor? This factor had in denominator little product between these two. Work toe These two Victor's We have five multiplied by three minus three multiplied by three. We have minus three, multiplied by seven plus City gives minus 12, divided by little product. Between Vito and itself, we have three squared multiplied by four, which is 36 minus the projection off the third column into the one one minus one minus 111 But the blood boy a factor. This factor had in the nominator the product between the third column and the one which is for you, plus four plus three plus seven plus one. She gives 20 divided by the product off the one and itself. She gives the flavor. These equals five minus. They have minus and minus one third. Then we have plus five plus three equals it. And we have here a minus on. This gives the four minus four. We have five plus three minus four. It gives the four. Then we have minus four plus three plus one. This is not for sorry. We have here five plus three, but applied by one third, which means five plus one minus four equals two. Then we have minus four plus four, which equals zero. We have minus city minus three multiplied by answer which she is minus one. Then we have minus minus four, which is plus four, which is zero have minus three plus one plus four equals two. And finally we have seven minus three, which is full. We have seven minus minus three, multiplied by onset, which is seven plus one minus four seven minus one plus four equals two and we have one minus plus one minus four equals minus two. This is the three. The second step is to normal, Iet these victors get in one as the normalization off. We won. This is we won. We divide by. It's normal. It's normal. Is the square root of five square root off. Five. We have one divided by a squared five multiplied by one minus one minus 111 Then we get to the same way we normalize V two. This is V two. We divided by square root off 36 which is six. Then we have one divided by half the blow it by 101 minus 11 Finally, we get the normalization off the three, which is here we divided by its normal, which is the square root of 20 which is one which is to but the blogger Square toe five. Then we have here Photo by 1011 minus one. Then we can construct the Q metrics, which equals in the first column. In the metrics we have in one we can we have one. Whatever school to five minus one. Whatever. School to five minus one, divided by square to five. One divided by square 25 and we have one by the by school to five. The second column is in tow, huh? Zero, huh? Minus half off. Third column is N three have one divided by square to 501 divided by square to five one Bible school five minus one divided by square to five. The final step is to get or, sir, the step is the first part of the question off the answer off The question stuck on the step. The fourth step to get our, which equals Q transpose multiplied by a matrix C. It equals one divided by square to five minus one, divided by square to five. It's wonderful. It was great to 5151 but it was good to five off zero off, minus half half on the viable square to 501 The Bible Scholar to five. One, divided by square to five minus one, divided by square to five. We multiply it by the metrics A, which is one minus one minus 111 21 four minus four to five, minus four. When a three 71 we can make the multiplication here toe get or or equals. The first element comes from multiplying. The first rule with the first victim, which is five. The valuable square toe five equals square toe. Five. The second. The second element here comes from the the first group of the boy by the second victim, we have tu minus one minus four minus four plus two equals minus five. The web squared five equals minus square to five. And then we must blow it with Victor. We have five plus four plus three plus seven plus one for the 20 divided by a squared five equals four or the blow by square to five. Then we take the second rope on. Then multiply by the first victor with second victory and third, we have one plus zero minus one minus one. Last one divided by two equals zero, Which is very logic because, or is a triangle of metrics. Then we have two plus zero plus four last four plus two, 12 divided boy to equal six and we have five plus zero minus three minus seven plus one equals minus four by the boy two equals minus two. Finally, we take the third row and multiply by the three victors. You have one minus warm. 10 minus one plus one minus one equals zero. Then we have to below zero plus four minus four minus two equals zero, Which is very correct. Finally we have for you plus zero minus three plus seven minus swamp equals it. Divided by the square root of five It divided by square root. Five we have here a little mystic. This is not it. The boy was called five. Because we have here this note this victor and three off Incorrect because we have calculated the norm. Wrong is square root off 16, which gives four. This means we have here half Oh, zero off off a minus huff. And this we have here off zero half off a minus off which gives us it divided by two which gives four here. And this is the answer. Mhm a second barred off the answer with solution and Q are here is the final answer of our problem.

Hello there. So for this exercise we have this matrix A. And we need to or to normalize the columns of this matrix. So that's why here I built in vectors the columns of these matrix A. So you want first column you two to the second one and you three to be a third one. And then we're going to apply the Greenwich mean procedure to obtain to our to normalize these columns. So and actually this is widely used to make the to obtain the QR decomposition of A. So let's start. So first we put in our Ortho, no we are going to construct an orthogonal set of factors. And here the hat the note unitary vector in this case these are not unitary yet. So first we build a set of orthogonal vector's and then we normalize them. So the first one corresponds to taking V one equals to you one. That in this case is equal to six to minus two and six. In the next step we need to take, we need to add a second vector that in this case is going to be V two. And that means transforming Youtube into a vector V two such that this vector V two will be Orthogonal to be one. And to do that we need to take me to equals to Youtube minus the projection Of you 2/2. So the projection of you to over two we want is defined as the product of you two would be one times we want divided by the square of the north. So let's calculate first the square of the norm B one. And in this case is just the product of Iran would be one that is equal in For this lecture equals to 80. And the inner product here. You too. The one is equal to 60 minus Yes, 60. So In that sense, director V two is defined as six, 2 -2, 6 minus 3/4. So here is the vector YouTube that is one one -2, 8 and -3/4 of 6: -26. That corresponds to the one. And we obtain the second factor that is of terminal to be one. That is one hell of minus seven minus one minus one seven. So we have four seconds Also an inspector. No we need to add the 3rd 1 inter set. So we want The two and we adhere three. So it's technically the same idea. We pick you three, we transform it to A vector B three such that this vector B three is are thrown up to be one. And to me too. So it's the same idea B three will be able to use three minus the projection of you three over the one and minus The projection of U. three over. So let's make this calculation to provide this. So the projection of you three Overview 1. We need The inner product of you three would be one we want divided by the square of the norm B one. This value we have already calculated here is just 18. So I'm going to right here just eight. Um uh for the projection Of you three over B two is the same U. Three over with to the product. So we need to calculate this uh square of the north of B two. So this for this specter Equal to 25 And the inner product of you three with me too Is equal to 10:10 And the inner product of you three would be one is equal two minus 80. Okay, so then we just need to replace here the values and we obtain that The Vector B three is equal To you three that's -11. Hi minus seven Plus the victory we want that is six 2. -26. And last one 2 50 It is 1150 one. Well -7 -1 minus one. Except. So the third vector equal to 1/5 minus two 14 14 two. Great. So after this whole procedure we obtain a set of orthogonal vector's. So we obtained set or orthogonal vectors. We want me to B three define asked Be one equals 2 six. Mhm 2 -26. A vector B two. That is equal to one half -7 -1 -1 seven. And the third vector it is equal to 1 50 minus 2 14 14 two. So these three vectors are are Thornell. So they form an orthogonal set but we require an Ortho normal set. That means that we need here to normalize them. We need to normalize directors to obtain unitary vectors. So that is a really easy procedure. We just need to divide the vectors by the corresponding arms. So this becomes unitary if we take everyone and we divide by the norm of the one. And in this case this unitary vector is the vector two, one, your yes One over to the square of five 31 minus one, three for a bit too the same over the norm. And this is equal to 1/10. I'm the vector -7 -1 -1 said And the last one yes calculated in the same way And it's equal to 1/10 times minus one, seven seven. Mhm. And these three vectors form an Ortho normal set. That is what we want at the beginning. So they this corresponds to the if we are in the process of calculating and or to normal the cure the composition of matrix, they will form the matrix. They will form the correspondent ah colors of the big of the matrix.

In problem line. We want to find an orthogonal basis for the columnist based off these metrics, the first system is to get the one one off these three victors. We will choose the first victor 31 by in a swamp. Three. Get the second victor. We use Graham ish method process. We two equals the second victor here, minus 515 minus seven minus the projection off the second victor onto the one. Then we write the one. It's just 31 minus 13 was deployed by effect. This factor, as in the nominator the dot product between the second victor and V one minus five, which equals directly. This is the product equals minus 15 plus one minus five. Minus 21 equals minus 40. Divided boy in the nominator. We have the dot product between few, um, and the one the product is three squared, plus one squared plus one squared plus three squared equals 20. These equals minus five plus six, which is one. We have one plus two, which is three. Then we have five minus two. Which three? Finally, we have minus seven. The last six when the seven plus six equals minus one. This is for Vito to get 33 The three is just the third victor here, which is one one minus 28 minus double rejection off the third victor in tow. The one M V two minus the projection on the third victor on the sack on Vito. We have the 2133 minus one multiplied by a factor. This factor, as in the nominator the dot product between the third victor. Envy too, Which is one one plus three minus six minus eight, which gives minus 10. Divided by the new product between we to which is one plus three squared plus three squared plus one equals 20 minus. The projection off the third off the third victor on to be one. Then we have the one we right. We won 31 minus 13 and we multiplied it by a fact. This factor, as in the nominator dot product between these two victors the third, the third call and the one which gives three plus warm plus two plus 24 which equals 30 divided boy, the dot product off we want and we want, which is calculated here. 20. These equals. We have one minus and we have here a minus plus half. Then we have minus three, multiplied by 1.5 equals 4.5, but with a negative side minus 4.5. The second value, one minus minus may have plus an email to global health, which is 1.5. Then we have minus 1.5 third. Value is minus two minus minus 1.5, which is 1.5. Then we have plus 1.5. Finally, we have it minus minus minus, which is minus half. And finally we have minus 4.5, which equals 1.5 minus 4.5 equals minus three. We have one we have minus to plus three, which is one, and we have eight minus half minus 4.5 equals three. Which means the fit. That's it that contains we want We have, um, 31 minus 13 31 minus 13 And the victory V two, which is 31 minus 13 No. Three. We one V two is 133 minus one and we three, which is three by the 3113 They said is the Ortho normal basis for the columnist base for the metrics given in the problem. And this is the final answer off our property.

To find the north orginal basis for the column space at this matrix. The column space was the subspace spanned by the column vectors of the Matrix. And so, essentially, we gotta spectra subspace That is the span of X one extra extra. And I've labeled them like this, right? And so I want to find a new basis for you want me to be three? That is the same. So that is the basis for the same subspace w the column space. However, I want this set of actors to be orthogonal, so v one not be too. Must be 031 dot be three. Must be zero. And so must be vey to dot B three. Okay, So clearly, we're gonna have to apply the Gram Schmidt procedure. OK to x one. Being this vector X to being this vector next three being this vector. So what does Gram Schmidt to tell us? It's a V one is equal to x one. Me, too is equal to x to minus a subtract x two dot v one Polar mont theme Magnitude squared of everyone. Lots of the one. And so this makes X two perpendicular to be one. And now for the three attic X three, I make it perpendicular to be one. Andi, I also make a perpendicular to V two. Okay, so this right here and is what I need to use formula that I need to use. Okay, so let's go to the next page. So we know what? Okay. So, basically, I want to find what V one is first, then find V two and then finally find V three. Well, V one is the same as X one, as we said. So I've written that down V two. So let's just focus on B two. It's this right comes okay. We need to figure out what this V two is is a vector. What? That what? The components are, by the way, have written them like this as rose. Just a safe space. Nothing else. Okay, so, as always, you should find what these guys are First, to make things a little bit easier. So what is x two dot v one? That's the same things. X two dot x one. Right. So it's three plus three plus five plus five. That gives you 16. What is V one more squared? That's one plus one plus one plus one. So that's four. And so this is the same thing is X two minus 16 by four times V one. So that's X two minus four times being one. So let's figure out what V to actually is. So that's three minus 3 to 55 minus four times one minus 1011 And so that's the same thing is three minus four minus three plus four to five, minus 45 minus four. And so that gives me minus 11211 Let me write that down minus 1121 point. Okay, so let me update. What we've got so far is one minus one zero's There, there. 11 is V one. V two is minus 11211 Now the question is, what is V three? So let's go back to what Gram Schmidt tells us. Be three is given by this. Okay, as always, let's figure out what these quantities are, first of all, so x three don't be one. So that's, uh, five, five times one minus one plus two plus eight. Okay, so five minus one plus two plus eight. So that gives me 14. Okay, x three dot ve to. That's so be careful that you don't use X to You need to use this guy right here. So it's minus five plus one plus six plus one plus hates. Okay, so that's, um Okay, so let me just check. Minus why was one plus six plus two plus states? Okay, so that's 10 plus two. That gives me 12. The square of the magnitude of the one. Well, we've already worked that out. That's just four square the magnitude of the two. So one squared, plus one squared plus two squared plus one speck plus one squared. That gives us eight. Okay, so All right. So I want to be careful not to run out of too much space. So the 36 x three minus 14. Divided by. It's a 14 divided by four. It was the one minus 12 divided by eight times. Me too. And so that gives me X three minus seven Harms. We want minus three. House beat you. Okay, so now let me actually figure out what this vector is. So now it's going to get a little messy, so this is going to be. So let me just write it out. So one line ist ones over one minus 11211 Yep. So this is extreme minus seven. Hearts be one minus three House. Me too. All right, this is what that is. So let's combine this into a single vector. We get five minus seven halves, plus three halves on. Yeah, just it's down to us arithmetic at this point, arithmetic that you should be careful not to mess up because it's really easy to mess up arithmetic. Okay, so that's five minus 21 plus 20 to minus five and eight, minus five. And so that gives us finally 3303 It's a 33 033 3303 Okay. And so it's a good idea to check that the one don't be too equals. We wound up the three because the 2.33 cause if this is Izzy, put zero. Because if this is false, then you made a mistake somewhere. Okay, All righty. So that's not conclusion that a North Oregon or basis for the column space is given by this. It's driven by this collection of actors. Okay, just is a note I sometimes switch between using square braces and round braces. It really doesn't matter. All right, so that's it. That's this question done. I'll see you in the next video.


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