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Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .$$\left[\begin{array}{rrr}{1} & {3} & {5} \\ {-1} & {-3} & {1} \\ {0} & {2} & {3} \\ {1} & {5} & {2} \\ {1} & {5} & {8}\end{array}\right]$$

$\left\{ \left[ \begin{array} { c } { 1 } \\ { - 1 } \\ { 0 } \\ { 1 } \\ { 1 } \end{array} \right] , \left[ \begin{array} { c } { - 1 } \\ { 1 } \\ { 2 } \\ { 1 } \\ { 1 } \end{array} \right] \cdot \left[ \begin{array} { c } { 3 } \\ { 3 } \\ { 0 } \\ { - 3 } \\ { 3 } \end{array} \right] \right\}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 4

The Gram–Schmidt Process

Vectors

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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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