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In Exercises 9 and 10, find bases for the column space of A and the null space of AT . Verify thai every vector in col(A) is orthogonal to every vector in null(AT )...

Question

In Exercises 9 and 10, find bases for the column space of A and the null space of AT . Verify thai every vector in col(A) is orthogonal to every vector in null(AT ) .4 =Ansnercol( AJ:null( AT )

In Exercises 9 and 10, find bases for the column space of A and the null space of AT . Verify thai every vector in col(A) is orthogonal to every vector in null(AT ) . 4 = Ansner col( AJ: null( AT )



Answers

Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .
$$
\left[\begin{array}{rrr}{-1} & {6} & {6} \\ {3} & {-8} & {3} \\ {1} & {-2} & {6} \\ {1} & {-4} & {-3}\end{array}\right]
$$

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.

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.

Problem. 11. We want to get on our signal basis for the column space for these metrics. The first step is to get the one which will be one of these victors. We will choose the first victor one minus one minus 111 The second step is to get veto, which equals the second victor. This the second column 214 minus four to minus the projection off the second column into If you want, we will. Belichick this victor in tow, the 11 minus one minus 111 Then we multiply by a factor. This factor has in the nominator. The new product between these two victors, which equals two minus one minus four minus four plus two equals minus five. And we have in the denominator the dot product. Between this victor and itself, we have one squared plus one squared plus one squared, one squared, plus one squared equals five, which equals to we have your minus one minus one, which is one two plus one equals three one minus one equal zero. We have four minus one equals three minus four plus one equals minus three. I'm finally we have two plus one equals three. For easier calculations, we can get V to dish. The two dish equals just the victor V two, but the skill as one third of it equals 10 one by in this 11 The third step is to get the three with three is just the third victor or the third column here, which is five minus four when a three, seven and one minus. The projection off the third victor unto V two or V two dish. Then way right V two dash from zero one minus 11 and we multiply it by effect. This fact this factor had in denominator the dot product between these two victors, which equals five minus three minus seven plus warm equals minus four, divided by the dot product. Off this victor and itself, which gives one squared plus one squared plus one squared plus one squared, which is four minus the projection off the third column in tow. We want we have the one three one minus one. No, the one is one minus one minus 111 and we multiply it by a factor. This factor has in the new minute the dot product between the third column on V one, which gives for you plus four block three plus seven plus one 20 divided by the dot product Off this victor and itself, which is one plus one plus one plus one plus one equals for you. One squared plus one squared plus one squared plus one squared squared equals five. This gives for you plus one and we have here minus four. We have minus four plus full. We have minus three. The last one minus four plus four on. We have seven minus minus one minus one minus four. And finally we have one minus one plus food minus four. Minus four equals six months. Four equals two zero toe seven minus five to finally, we have one minus 10 minus four. No, it's here minus. We have here plus one plus one minus four, which she gives here to minus four, which is minus two. We can't get it toe one half or not. Then that's it. Which contains we want, which is one minus one minus 111 And we to which is 10 one minus 11 And the three, which is 20 Tau minus two, you said, is the orthogonal basis for the Economist based for the metrics given in the problem, and this is the final answer off our problem.

Hello there. So for this exercise we have these three factors, the one between the three. And we know that these three factors form an orthogonal basis for our three. So let's we call this set of these three elements. Ask. So given the MSM is an also no basis for our three. We know that in particular this electric view that this an element of our three can be written as a linear combination of the basis elements in this form. So given the linear combination of you of the basis elements to form you, we need to give what is a correspondent coordinate factor you in the basis. S Well, basically you can construct this vector very easily won. Jihad, Mhm linear combination of the basic elements. So you need to be just the coefficients in this case 0 -2/3 on minus one third. So the coordinate vector will be zero -2/3 and minus one third.


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