Question
5 . Consider the subspacesU = span{ [ ~3~5 ] , [5 ~6 ~7 ]}andW span { [ =1 ~1 5 ] , [ 11 -16 2 ]}of V = Mix3(R)_ Find matrix X eV such that UnW = span{X}:
5 . Consider the subspaces U = span{ [ ~3 ~5 ] , [5 ~6 ~7 ]} and W span { [ =1 ~1 5 ] , [ 11 -16 2 ]} of V = Mix3(R)_ Find matrix X eV such that UnW = span{X}:
Answers
Find a basis for the subspace $W$ of $\mathbf{R}^{5}$ orthogonal to the vectors $u_{1}=(1,1,3,4,1)$ and $u_{2}=(1,2,1,2,1)$
This problem was given the metrics I has to determine. Finally, integer a such that no, a record space of art to decay and told the ladies and the K K K senses. No, actually, we're looking for a solution off this form. So a is a four by three mentions to test for Rose and three clothes. So for right? Three. And if you want a most split, this with a vector, director should have the same size, same number. So the columns up they should be called the rolls off. He's a writer, so this should be t I want. So it means that hey should be free. So then no, A and R Jew can be. Let's say we have another Victor X and let's say R B is a eyes column a again, a cz or rectory. So rumors we know that extra be three by one. It's a number Collins off a is equal number off rolls of X sign on the metrics. Speed will be up to size for by one. So it means that K should be four and then hold A and were subspace in our before
In this problem, we're s to find value case like that. No, it is a subspace in our to decay in the column. A subspace in our cake so hard today. No es. So as you can see, our has one roll. And 12345 colds. Right So than a ISS one by five. Um, makings. So since this no way. We're looking, uh, percent of recreational perform eggs, is it zero and now is on my eye and no. Yeah, we're looking for eggs here. That X should be open for five. Fine. One or five by one reason is the number of columns and a should be cooking number Rose in are, uh, eggs majors. So from this case should be five then l A is the subspace pain or fine part. Be cold, eh? So this time work creation looks. There's also perform exit B. So a CZ again one, my fine. Now we know not be a cz fine by one from front, eh? So since the number of Poland's and a mixes with the number rose in eggs, we want I have a problem and the resulting medics will be up to size one by one. So from this Okay, Is these what I call a maze of subspace and or
She watched him. Were given a vector V. V has components 1234 six. Were given a subspace W. And rested on the projection of the anti W. In other words the vector W. In the subspace W. That minimizes the length of the minuses allergies. In part A W is the subspace of R. five standby. See the vectors you won. Which has coordinates 1- 1- one. I thought dreams like this. Yeah. And you too with coordinates one negative 12 negative 11 Lucas. Yeah. First of all it's pretty clear that U. one and U. two are orthogonal. So all we need to do is calculate for a coefficients. Our first fourier coefficients. C one is the inner product of V. With new one over the inner product of you, one with itself. And this is one plus four plus three. Yeah. I mean it's like yeah plus eight plus six. Set Yeah. All over one plus four plus one plus four plus one. And want that once you. Which is yeah. Okay good. I saw it speech Yeah. Just 22 over. Okay. Think 11 or two. And the other four EKO efficiency to this is the end product of V. With you. Two over. The inner product of U. Two with itself. Well sure. Okay. Yeah This is 1 -2 plus six. Girls trip. Seems like -4-plus 6. This is over one plus one plus four plus one plus one. And then the best part is that great. This is that's how that works. She just starts pissing also let's seven over and 7/8. It's okay. All right. And therefore we have the projection of our vector B onto the subspace. W. Oh This is C. one times you won enter Plus C. two times you too. Which is two times one 2121. Diarrhea. Listen yeah plus 7/8 Times 1 -1. 2 native 11 This simplifies to mm 1/8 times three. There's got to be something 23. You make. Yeah 25 30 25. 23 grab something. Mm. Mhm. Then in part B we're told that you know what I said the subspace W. So is the span of the two vectors V. One which is 1 to 1 to one And V two is 1015 -1. Okay. Yeah. Fine. Yeah. Was ladies? He was unlike the previous parts. It's not clear if these are in fact orthogonal. All right. They too much iron. No. Do I just however if you calculate You'll find that V one and V two are in fact not orthogonal. Yeah. Your dog product is not zero. Yes. Okay. That was such both work. So it's a hope. Yeah. Look you're ralph stuff just you know, I also hadn't been so nice. Yeah. Yeah, we got there help. I I'm right. Mhm. Icing this. Yes. So first you want to find an orthogonal basis. So they're not orthogonal. That's what I mean. They are linearly independent. The nut or fog find an orthogonal basis. I'll use the gram Schmidt process. Yeah, there's Edison involvement. So I'll take W. One to be the same as vector V. One which is 1 to 1 two, one. Yeah. Uh And define W. Two. Well, let's find first V two minus the inner product V two with W one Over the entire product of W. one with itself times W. One Which is 1015 -1 minus In a product of V two FW 1. This is one plus zero Plus one plus 10 minus one. All over one plus four plus one plus four plus one times staff 12121. This is 101 5 -1. So I had two different c minus straight eyes left. 11 11th, Which is just -1 times 1 to 1 to one, which reduces to the vector. Wait zero negative two, zero. Yeah, three negative two. And so I'll take our second vector W two. Really? Wait. I've really been. This is um when I said it last week, I multiply through by negative one And then we get zero two, zero negative three, two posted the screenshot. My gram Schmidt follows that Set W one. W 2 is in orthogonal basis for our space. W Yeah. Don't know what you mean by real legend. There. The head system rape Children on the right, okay. Here's a few minutes. And to find the projection of the end of w the space, let's find the fourier coefficients U n c two. C one is the inner product of vector V On two. W 1 over the inner product of W one with itself. That's true. That's his job. This is actually the same as before. This is also too Hansel and Gretel story. Then the inner product Of me with W two over the product of W two is itself. Well, this is gonna be different. Get out. I was like you, so yeah, I know. Well, this is uh zero for zero. Yes. Right. Yeah. Mhm. Yeah. Mhm. Okay -12 Plus 12. I said believe all over. Yeah. Okay. Zero plus four plus zero plus nine plus four. This is pieces of, like simplifies to Okay, for sure. Mhm, 17th. Mhm. Yeah, it was. And therefore the projection vector V onto the space W. This is C one times W. One plus C. Two times W. Two. Thanks. And so this is two times 1. Two. 1 to 1. Yeah. Plus 4/17 Times W. two which is 0-0 -3 negative 32 being cannibals. And this simplifies to, mm. Okay. Mhm. Yeah. 1 17. Yeah. Crash. It's like, yeah. Well I don't it could be worse as if yeah. I think this is actually 38. You're all eating there. Everyone. It was right to kill jews and It's 117 times 38 then Yes 76 34 Forest Whitaker 56 and 42. Yeah. On the Chelsea.
So we want to write. Uh, why as a sum of vectors one a w and one orthogonal of W where w is just, uh this pan of all these use. So first, to find the one N w we need to project why they were on Waned of project. Why North, Ogden Lee, onto the span of you want you to endure three. So to do that, we'll take our projection formula and project. Why? Onto you won. You too on you three. So let's find all these DA products off to the side. Why? I thought you want iss three plus four. Plus they're a minus six, which gives us one. You want dot product with itself. Make it one plus one. Hello, zero plus one. So I get three. So those first coefficient is 1/3 Next. Uh, why that product with you too gives us hurry. Close zero most five plus six. So eight and six gives us 14 and you, too dot product with itself. Look it 1.0 close one plus one. So three again. So our second coefficient iss 14 over three. Finally, uh, why dot product with you three we get zero minus four. Close five minus six. Negative. 10 plus five gives us minus five and you three dot product with itself again, we get zero plus one plus one plus one. So three. So our last coefficient is negative. 5/3. So now we're going Thio multiplies coefficients into all the inspectors. So 1/3 time's you won gives us 1/3. 1/3 0 You could have 1/3. Ah, second victor, 14 3rd multiplied to you too. Gives us 14 3rd everywhere. Um, except a zero in the second place. And lastly, negative 5/3 multiplied into you three. So we have a zero 5/3 minus 5/3 and 5/3. So we're gonna add all these up. 1/3 plus 14 3rd ISS 15 over three, which is five 1/3 and 5/3. Gives us 6/3 or two 14 3rd Minus 5/3 is 9/3 or three. And lastly, 14 3rd plus 5/3 and 19 over three minus one over three. Gives us 18 over three or six. So our production vector that's in W is 5236 So we need to find the one that's 4000. All the w. So to do that, we're just going to take the vector we want to get, which is why subtract. Why? Hat? So we have 3456 minus that vector we just found, which is 5 to 36 So this gives us negative too. Two, 20 So therefore, um, we can rate why? As a song, Uh 5236 plus minus 22 to go.