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Consider the subspace V of 93" given by V = span{v,,6,,6, where ", = (1,0,0,-1). uz =(-4,1,1,O)and #, (8, 3,6)- It can be shown that A={u,,u,,us basis for...

Question

Consider the subspace V of 93" given by V = span{v,,6,,6, where ", = (1,0,0,-1). uz =(-4,1,1,O)and #, (8, 3,6)- It can be shown that A={u,,u,,us basis for Applying the Gram-Schmidt process to in order t0 find an orthogonal basis B = for yields (1,0,0,-1) and Vz = (-2,1,1, 2); hence;, find v, Find corresponding orthonormal basis for V

Consider the subspace V of 93" given by V = span{v,,6,,6, where ", = (1,0,0,-1). uz =(-4,1,1,O)and #, (8, 3,6)- It can be shown that A={u,,u,,us basis for Applying the Gram-Schmidt process to in order t0 find an orthogonal basis B = for yields (1,0,0,-1) and Vz = (-2,1,1, 2); hence;, find v, Find corresponding orthonormal basis for V



Answers

Let $W$ be the subspace of $\mathbf{R}^{4}$ orthogonal to $u_{1}=(1,1,2,2)$ and $u_{2}=(0,1,2,-1) .$ Find (a) an orthogonal basis for $W$ (b) an orthonormal basis for $W$. (Compare with Problem $7.65 .$ )

Hello there. So for this exercise we got these three factors B one The two and B. three. And these three vectors is a super space in our four in this case. So we need to find a basis for the orthogonal complement of this based off of you. So how to build this whole? Let's recall the definition of the orthogonal complement is equal to the vectors X. On the space. In this case four such that X. Is an internal to the bacteria V. For all the the space of you. And this condition here is they have anticipated. X. V. Is equal to zero for all the. No. So the point is that we can obtain your final compliment. Just be considering the generators of this space. And what I mean is um considering B. One, B. Two and B. Three. So let's be a generic point X on the compliment. Then this implies that these vectors satisfied that x. v. one X. Movie too and six B. Three. All these are equals to zero. Because if this vector X. And any any vector X. In the compliment will be also known to the generators. Actually there is a generic vector of the form X. One. Uh huh. X. Two X three 64 So this becomes a system of linear equations that have matrix representation that corresponded just built the the vectors as columns of this matrix here. So I mean put in here one for 5 2 days be one 2130. This v. two And the 3 -1 three 21 This vector is multiplying X. One X. Two X. Three X. Mhm equals two 00 Okay so we got this system of linear equations and we can, what we need to find is the a solution for digital space orginal space of these interests here. So for that we're going to reduce that metrics to the Russian form the whole system. So we obtain the following we obtain 101 -2 over seven 011 for over seven And 000 zero times X one x two X three X four. And there is the cost to the 000. Okay. So from this you can observe that here we got a free variable that corresponds to explore. And here we got another free variable that response to X three. So what does mean is that the null space he generated by two vectors. Which is equivalent to say that I love you will be generated will be dispatched and of two vectors. Let's call off for one of two. Okay? Um so let's find those vectors. And those vectors will be the will be obtained by calculating the general solution for this system. So the general solution, If X three and X four are free bibles, that means that X three could be close to t the next four. We're going to give the value of our So the general solution X. Is equal to t -1 -110. Lust har And the vector to -40. So so you can observe here that the and any victor in the any vector X. In the compliment Is generated by two vectors by the linear combination of two vectors. These two vectors corresponds to offer one. Mine is a warning one, 10 and alpha two Which in this case is equal to two minus four, zero and seven. So these two vectors expand the whole space dog and even more they are linearly independent. So two vectors selectors that spanned the space. So we know that the space the arsenals compliment will be this pond here of the one of two and Alpha one Alpha 2 are linearly independent. Then the basis for the compliment will be The vectors all for 1/2.

X one is the CO 21201 and it's too is you. Go to to 110 x three Issue Come to 1021 So egregious Grinch meets we have Everyone is Nico two. X one is equal to 1201 Ines If we so the second it to it will be two of the X to minors, thanks to me. One into the one square you want so we can play with X to be one there is should it come to pass two plus zero Prospero, this will cut your four into the one square. It's ICO, too. 12 square pass one. It's Nico to six. So over be to issue going to execute its 211 barrel miners to over three one to their one. So here the first week it's to minus two of the three four with three, then one miners forward. Three will be my next one Street over towards me Into one ends in my news tourists. Then we have three. It's the culture extreme miners next 331! You want square in the one minus extreme. Me too, to square me too. So the company said that unit protects extreme in the one so we can get is 10 plus zero plus one Here, we'll get to into we compute extremity in V two. They're your kids over three That's zero to miners to oversee. So this will give you this will give you eight oversee into the known square off to give you who Where those one always minus one squared. That's one cost minus two over ST Square This well, ico too. Is this where 16 pass one plus nine plus for overnight Where do 10 over a suit. So over this very insecure to 10 to one miners two over there. And be one. It's 1 to 01 instant minders. It's over three in the 10 number speed. The thing is, we'll always be minus 1/3. One money to So here you can have one go through one money for number three 12 year one my overwork flying for ST minus one or 31 minus two. So this will you, you one minus one minus 15/15. We'll give you minus two of by in there minus minus to oversee over 15. We'll give you two are fun and then you have to minus forward. Why give you 6/5? Then you help one minus one Nowhere Street love. It's over 15. Give you sorry. Thank you. 6/5. So over. Or but you're also gonna basis will be the one in the two and the three. 6/5. 6/5.

Right. Yeah. Were given to Victor's. You won in our four. This component 1 -2 34 Another vector in R. 42 with components three negative five seventies. That shit rules. It's a net ST anger and saying documentaries. And we have to find a basis of the south's based W. Of our four. Orthogonal. Should U. One U. Two. Good sound. That's I saw the sea. Well, describing something he's looking inspired and it's in supremacists. Oh in a prison. Are you insane? Quit mentally. We know that if the lies on this basis curtains well let's say he has the form uh X. Y. Z. T. Then V. As in W implies that the enter product of V. That you want any other parts of the with you too. At the 200. Using the.ble this out. This implies that uh X -2. Y plus The limousine plus 14 people, zero and three X minus five. Y plus 17 plus 18 equals zero. You obtain the system of equations right? We'll simplify this system. Has tracked three of equations. One from equation too. So it's still an equation one X -2 wide. Plus there is the Plus 40 people zero. And then You have negative five Plus six is 1 y. Seven -9 is -2 Z. And then you eat -12. There's nearly 4. 2 equals zero. Much shittier. Yeah. Now in this new system we CFC three variables take for example uh two equals zero. And let's take you to be one. Then from the second equation it follows that Y is equal to two. And from the first equation it follows that X is equal to one. Therefore one of our vectors. Step it up to Slur Is 1- 10. It's like to find another vectors well Because there are two non yeah linearly independent vectors. You know that the subspace W. Has mentioned to you to find another doctor. And now, instead of zero, let's take to you to be one And take Z to be zero. Yeah. Then it follows that. Why is he going to four AmeriCorps? My first equation x equals four as well, that they listened to it? They said the podcast is they read the front And so we get the .4401 and therefore the set, wasn't it 1210 for for 01 forms a basis for the subspace double risk. Fresh. What is this place?

Right I mean, Were given a subspace you of our 4? Yeah. This one spanned by the vectors. The one with coordinates 1111 there, V two With coordinates 1 -1- two. And the three with coordinates 12 -3 -4. Yeah. Yeah. Put in the Yeah. And in part a were asked to apply the gram Schmidt algorithm to find an orthogonal and an Ortho normal basis for you. Yeah. So as part of gram Schmidt, I'll take our first basis factor. Nice. W one. This will be the one which has coordinates 1111 to find your second basis factor. I'll calculate the tu minus In a product of V two with W one over data product of W one with itself time W one. This is equal to Sector one, negative 1-2- The Enterprise of V two. W 1. This is 1 -1 plus two plus 2. Yes. Over one plus one plus one plus one times some sun 1111. So this simplifies to 1 -1-2-. This is four Over four, which is one times 1111. Yes. Which simplifies to zero negative two 11 It's like you know I've never chance on him. And so I'll take our second basis vector W two, B 0 -211. Just one time 4 years old. Now there is one more vector to find W. Three. To find this, let's find V three minus the inner product of V. Three with W. one over the inner product of W. one with itself times W one minus theater product of B. Three with W. Two Over the inner products of W. two with itself times W. Two. Mhm. It's all to stop doing kind of contact everyone next And this is equal to 1 2 -3 -4 The projection of the three on 2. W one is one plus two minus three minus four. Over one plus one plus one plus one times 1 1. 1. 1 minus senior product of V three with W two which is zero Easy minus four minus three minus four. That Zach braff. This seems to over at least the entire product of W two with itself which is zero plus four plus one plus one times first. 0 -211 Davis. Oh it Which is equal to 1 2 -34 Sinus Messages 3 -304 positive 4/4 which is one Times 1111 a minus. Okay this is all negative 11. So plus 11 over six who are they Times 0 -211. Yeah. Mhm. I just need more in close. Mhm. This simplifies to one plus one is 22 plus one is three minus 11 3rd. Oh we stop raping me, hub negative three plus one is negative two plus 11. 6 and four plus one is five plus 11 6. It's clever. That's good. Not long ago. And so simplify I'll multiply through by 6 to get whole numbers and we get I'll take our basis vector W three to be two times 6 is 12, 18 -11 is seven negative 12 plus 11 is negative one and 30 plus 11 is Yeah. Uh This should actually be 12 negative three negative four then this is correct. Yes. Yeah. Talk. Yeah chris concerns think for so this simplifies to Celebrity insider said this is negative three right plus 11 6. And so this is 12. We know that seven negative one and that Arnold I made a mistake here, This is uh 18 -22 is negative force. And this is negative, 18 plus 11 is -7. 12, negative four, negative one negative seven. This is our third basis vector W three. And by gram Schmidt, it follows that right vectors W one W. To give you three. These isn't orthogonal basis for our subspace. You right? Yeah. Wayne, Yeah. Plan Ortho normal basis for you will simply normalize each of the vectors in this set. The norm of W one squared is one plus one plus one plus one, which is four. Like why is the norm of W two squared, right? Is zero plus four plus one plus one. Was like all that gold, which is six. And then the norm of W three squared is 12 squared, which is 144 plus 16 plus one plus 49 that trinity I was. Which is Yeah, but I guess This is uh 61 60 plus 50 which is 210 asking God that's who's the villain. Yeah. And therefore we get the new vectors U. One which is one of the norm of W. One which is one half times W. One. So this is the vector one half. One half, one has one half. No that was W. Tracy. Ellis ross you too. This is W. two divided by the norm of W. two. Yes. Which is one over root six times 0 -211, the last last And the Third Unit Factor You three. This is W. three over the norm of W. three. Just one over the square root of 210 times 12. Yeah -4 -1 -7. It was all like heavily because W 12 W three were north diagonal sets. It follows that beset U. One U two. U three is an Ortho normal basis for our subspace. You what then? In part B? Yes. We're asked to find the projection of vector V onto you. Where the is the vector was components 12 negative three positive force. Really? Oh No one has black city local to do this. We need to calculate the fourier coefficients. Yeah. So my first foray coefficient, this is the inner product of V. With let's say The simplest one to choose would be w. one Over the inner product of W one with itself more which is uh one plus two minus three plus 4/1 plus one plus one plus one. Which is work one. Yeah. 2nd 48 coefficient is in a product of the with W two of the product of W two itself. This is 0 -4. Mhm -3-plus 4 over 0411. This is negative 3/6, or negative one half. So. And then the third fourier coefficients C three is the inner product Be with W three Over the in a product of W three with itself. Yeah, he was like, this is 12. This could mhm. Right, negative eight. Three. Yeah. No. And he brought her on as minus 28 over. Uh Well, this is 210. Yes. Is equal to right mm negative 21 over 210. Which is negative 1/10 brought up since really rusty. And therefore the projection scandal uh V onto the subspace. W This is C1 times W one plus C two times W two plus C three times W three. Right? And this is one Times 1, 1, 1, 1. 100%. I could be I minus one half. Yes. 0 -211 sorry Uh -1. 10 Times 12 -4 -17. It's our -7. That makes me madder. Left station. Yeah. Just going. Probably. Yeah. MS reduces to yeah. You're serena 1/5 times negative. One girl. Yes. 12. But instead of big nose, you swap. Yeah. Three. Don't do the math and six racist. Yeah. And all this shit.


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