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[M] Use the Gram-Schmidt process as in Example 2 to produce an orthogonal basis for the column space of $$A=\left[\begin{array}{rrrr}{-10} & {13} & {7} & {-11} \\ {2} & {1} & {-5} & {3} \\ {-6} & {3} & {13} & {-3} \\ {16} & {-16} & {-2} & {5} \\ {2} & {1} & {-5} & {-7}\end{array}\right]$$

$\left\{ \left[ \begin{array} { c } { - 10 } \\ { 2 } \\ { - 6 } \\ { 16 } \\ { 2 } \end{array} \right] , \left[ \begin{array} { c } { 3 } \\ { 3 } \\ { - 3 } \\ { 0 } \\ { 3 } \end{array} \right] , \left[ \begin{array} { c } { 6 } \\ { 0 } \\ { 6 } \\ { 6 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { c } { 0 } \\ { 5 } \\ { 0 } \\ { 0 } \\ { - 5 } \end{array} \right] \right\}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 4

The Gram–Schmidt Process

Vectors

Johns Hopkins University

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Oregon State University

Harvey Mudd College

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Okay, so we're given this matrix, and we need to find a North organo basis for the column Space of a. So, as we said before, this really just pulls down to applying Graham Schmitz to the column vectors of the Matrix. Right, Because the column vectors of the Matrix spun the call them space, right? And so we want to find a new basis v one v two v three before that is a basis for the same subspace but his orthogonal. And so we just need to apply Grand Schmitz to the column vectors. All right, so let's do that. Okay, so V one first of all is equal to X one, right? I'll write this done. The bill later V two is equal to X to minus x two dot ve won over one of those people in squared times. Do you want? Okay, Andi, Then I'm not gonna bother writing down v three and before yet, we're okay. So if we figured out what V one is, right, it's the same thing as X one. We want to figure out what V two is, and then we'll move on to vey three and before, so as always. We need to figure out what these guys are first. So x two dot v one, which is the same thing as X to doubt X one that's is so it's minus 1 30 plus to minus 18 minus 256 plus two arriving just taking their dot product that gives me minus 400. This is the same thing right here as 10 squared plus two squared plus six squared plus 16 squared plus two squared, which gives us 400. And so v two is equal to X to minus minus 400 over 400 times the one and so that's equal to the vector. Claritin 13 minus 16 1 plus minus 10 to minus 6 16 too. And so that gives us. And I'm writing these, by the way, as a CZ rose just to save space in my writing up. Okay, that's all. So V two is +33 minus +303 Let's go to the next page. So V one is minus 10 to minus six 16 too. On we found that V two is 33 minus three 03 Okay, so the 1st 2 are down. What is V three now. Well, V three from the ground. Sweet procedure that is equal to x three minus extra dot V one divided by B one squared times. We won minus extra Dr B two, divided by magnitude V two squared come to me to again. I should put arrows on all these, but that would make the notation really messy. Okay, so let's figure out what these coefficients right here are so extra dot V one. That's the 1st 1 that one minute figure out. So that's seven minus 5 13 minus two minus five. Dotted with minus 10 to minus six. 16 too. Okay, so that's minus 70. Minus 10 minus 78. Minus 32 minus 10. And so that gives us minus 200. So extra dot V one is minus 200. This right here. We already figured that out, and that was equal to 400. Now, let's figure out what this isn't what this is. Um, we'll do this over here so extra dot V two is equal to it's a seven minus 5 13 minus two minus five. Well, they say you shouldn't do this as fast as I'm doing it. um there are a lot of room for errors if you're not careful. So that is this. And that gives me minus 48 seven times 3 21 minus 15 minus 39 minus 15 usually minus 48. And so, minus 48. Just trying to save some space here. And then we're gonna figure out what? Uh, well, we've already got feed too. Minus two beaches squared. So Okay, so X three dot V two is equal to, um minus 14. That was it. Okay, so move over here. So the three is equal to x three minus minus 200 divided by 400 times V one minus minus 14. Over 36 times. Meet two and simplify this We get Yep. This 4/3 meets you. Now, let me write out the components, cause I won't actually figure out what the vector v three years, right? What the components are. So that's this. Ah. I do apologize for going between back and forth between the square braces notation and this horrors, you know, writing his rose. But it really doesn't matter. Okay. Anyway, so seven minus 5 13 minus two, minus five. Minus 10 to minus 66 Dean two, Right? And then V two was three. I'm just reading it from here. 33 minus 303 And so that when you combine it, you can check. I'm running out of space, so I just gotta write it down. So seven minus five plus four. That's gonna give you six minus five plus one plus four. That's gonna give you zero 13 um, minus three. That's 10 minus four. That's good. TV six again. Minus two plus eight. Six. Then that's just six there, minus five. Plus one. That's minus four. Plus four. That zero. And so we figured out what, the three years with six comma Zero on my six cover 6 to 0. Okay, so let's updates now. All right, so this was minus 10 to minus 6 16 to and this was 33 negative. 303 And this waas six 066 zeros. It's fine. Okay, finally, we gotta figure out what the four is so before is equal to x four minus x four dot be won over magnitude of you one squared times the vector V one minus x four dog V two over magnitude of you two squared times. Me too. Minus x four dot fi three over magnitude. Be three squared times V three. Okay, so, as always, we gotta figure out what the coefficients are first. So x four docked V one? No. What is that? Minus 11 3 minus 35 minus seven. All you gotta do is look here. And that's why I keep track of all the vectors appear could just read them off. So the one that's minus 10 to minus 6 16 too. So that gives you 110 plus six plus 18 plus 80 minus 14 on that's gonna give you 200. Okay. All right. So Okay. So x four don't be too. That's the same thing as so minus 33 right? Looking at me, actually, write it out. Minus 11 3 minus 35 minus seven. Okay, this is running out of space here, so let me write it like this, dotted with reading this week to hear 03 So that's AA minus 33 plus nine plus nine minus 21. And that gives me minus 36. Okay, so I'm gonna just go into the next page. Um, all right. So, yeah, I'm gonna go into the next page. So explored up the one we figured out was 200 export of each year we figured out was minus 36. Thanks for dot V three. You can check that. That's minus 54. Okay. And then finally. So, um, so we got this already without this already? What is V three? Much of 73. So this guy right here, that was 400. This was 36. And then this We'll go back to. So that's six square plus six squared plus x squared. You could see that from over here. Right? The six squared plus six were to six squared. So then that's equal to 108. Yeah. So s o x four dot V three. That's equal to minus 54 and then one jealous. Okay, Squared is equal to 108. Right? Can the whole point of me doing all of this? So all this working over here, this is so that I can just substitute directly into here so that I don't get into too much of a mess. So now that I know these, I can figure out before Savi Fours x four minus x four dot me Warn over this Just write downs. Yeah, so let me just plug in the coefficients. At least these scaler is right here. So that's it. Minus 36. Divided by four. Nope, That's not right. Sorry. 200 divided by 400. It's the one minus minus 36. Divided by 36 times. Me too. Minus minus 54. Divided by 108 10. 33. Well, that simplifies to this, right? No, go back over here right again because I'm kind of running out of space and account easily. Erase. Um, you know, you should have this right here in the back of your mind using that substituting. And you gets these guys here No. 660 and you can check. So minus 11 plus five plus three plus three, that's going to give you zero. So three minus one. That's two plus three. That's five minus three plus three, but zero minus three. Plus three. That zero again. Five minus eight. That's minus three. But then plus three zero minus seven, minus one. That's eight plus three. That's five. Uh, sorry, minus five. And so v four is this? And let me update we have here. So that's oops. So 0500 minus five and so on orthogonal basis for the column space, which we were asked to find, is given by this right here. And you can check that this is indeed a north organ. All set on where dunk.

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