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.