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(10 points) Show that the set {u1,4z,u3} is AI orthogonal column vectors U] basis for R% 3,0). 42 where the (2 2 Kr1) ad 43 7(5,23.1) as a linear 1,4); Also express...

Question

(10 points) Show that the set {u1,4z,u3} is AI orthogonal column vectors U] basis for R% 3,0). 42 where the (2 2 Kr1) ad 43 7(5,23.1) as a linear 1,4); Also express x combination Of the orthogonal set {ue} k 2,3.

(10 points) Show that the set {u1,4z,u3} is AI orthogonal column vectors U] basis for R% 3,0). 42 where the (2 2 Kr1) ad 43 7(5,23.1) as a linear 1,4); Also express x combination Of the orthogonal set {ue} k 2,3.



Answers

Show that the vectors are not orthogonal with respect to the Euclidean inner product on $R^{2}$, and then find a value of $k$ for which the vectors are orthogonal with respect to the weighted Euclidean inner product $(\mathbf{u}, \mathbf{v})=2 u_{1} v_{1}+k u_{2} v_{2}$. $$\mathbf{u}=(2,-4), \mathbf{v}=(0,3)$$

Okay. This question wants us to show that these two vectors form an orthogonal basis for our two. And then we want to write the vector X as a combination of our new basis vectors. So starting out, we just need to check that these two vectors air actually orthogonal and you won dotted with you too would be two times six plus four times negative three giving us 12 minus 12 which is zero. So now we know we have a basis because they're orthogonal. So that means we can write any vector in our two, including X as a linear combo of thes vectors. So it would be some constant Alfa times are first vector plus some constant beta times. Our second vector or filling everything in now nine negative seven is equal to Alfa Times to negative three lots beta times 64 And now we can write Elin your system out of this. So we get to Alfa plus six Beta is equal to nine and negative three Alfa plus four beta is equal to negative seven. So now we can just go ahead and solve this. So from the top equation, we get Alfa plus three beta is equal to nine halves so out, so Alfa is equal to nine halves minus three beta, so we'll plug dead in down here and then distributing out here we get negative. 27 halves plus nine beta plus four beta is equal. That negative seven or 13 beta is equal to negative. Seven plus 27 halves or 13 beta, is equal to 27 minus 14 over to giving us 13 halves, so beta would then just be one half. Then we plug in one half in up here so Alfa would be nine halves minus three halves, which is six halves or three. So Alpha's equal that three and betas equal to one half, so therefore X would be equal to three you one plus one half you, too.

Hello there. So for this exercise we got these two vectors vector U. And vector B. And first we're going to check these two vectors using the Euclidean norm. They are not our tone. And then we're going to use a different in their product that is awaited in their product. And we're going to determine the value of K. That makes these two vectors or thought. No. So let's start with your clean case. So that is just some of the multiplication of each of the components of the vectors. That means to minus three. This -1 and clearly is different from zero and therefore you and we are not not Firmino using nuclear, you know, But now let's see what happened in this case. So in this case we're going to use the weighted ignorant product. So that is a really fine here. So it's going to be too times two plus key times minus three. So this is equal to 4 -3 k. And we need these vectors to be or thermal in the space, so we need to find the value of K and that is the K is a cool two four thirds. So if K is equal to 4/3, then you and we are phenomenal, but using the weighted inner problem. Mhm.

Hello there. So for this exercise we have this back to you and the soup spaced of you defined by the spine. Of these two vectors V one MBT. So technically these soup space W. Corresponds to a plane. We need to calculate the projection of this victory onto this place. Generically you can complete the projection of you into any said or subspace as the sun. All vectors in W. Of the England problem. The if you divided norm is part of the norm of the factors time to speak. So basically what I am saying here is the I am projecting you to each of the generators of the subspace. So in this case it means that the projection of you onto this of space of U. Is equal to the prediction of you On the Vector B. one. That this is fine by this formula. Plus the product of you, B to the responsible too projection of you wanted the vector. Uh Okay so we just need to calculate each of the components focusing. So let's start by calculating the products. So you be one is equal 21 plus three. Physicals to four. The inner part of you would be too is equal to and then the norm the corresponding norms the square of the time. So the square of the norms why is equivalent to say is equals to say the inequality one he did so And this is equal to one was war one which at the end become sex and the same for you too. It's just four was one is equal to five. So here we have all the components required for targeting the projection of you and w. So we obtain the projection of the victory. You innocent space of you complain is equal 246 is actually two thirds times one minus 21 plus the fifth times two. Why? So basically what I did here was taking B2 square is equal to five. So that's why we have these five here during the you would be two is equal to do. So we have here two fields and same here the the square The norm is equal to six. The Interpol to the cost of four. So it has become for sex and days after simplifying is the 2/3 that we have here And here. We have B one and B two. Just applying, replacing the said that. We have calculated here on the phone of the protection. And as a final result we obtained this is equal 2 1/15 Times are back for 22 minus work team. So this response is a projection of you on to this point of view. And now we need to calculate the alt component of U. With respect to this subspace up basically, that means it's obstructing the protection you mind as a projection of you in this space or playing this. After placing data is equal to one 03 minus the projection that we have already calculated here. 1/15, two -14 and 10. And the result of this is equal two one over 15 times minus seven, 14 35.

So in the given question we have two vectors which are given us seven I minus 14 G and to i minus G. And we are told to show that these victims are authoring right? So orthogonal vectors are victims such that the angle between these vectors is equal to by by two Or 90° right? Which means these vectors are perpendicular vectors. So what this means is when we take the dark product of these vectors, we will get the magnitude of eight times the magnitude of b times cost. Theta. Where theta is paid by two. So costly to is equal to paper cost paid by two is equal to zero, which means the doctor Director of Portugal Directors is equal to zero. So then we can find if two vectors are orthogonal by taking their dr direct. And if it is equal to zero, we can say they are author colonel. And if it's not we can say they are not orthogonal vectors. So the vectors that are given over here are seven i minus four J. And do I bless G. Right? The 2nd victories to a plastic. So now let's take the daughter of this vic test. So we'll take the dog protect by Taking the product of the components, the magnitude of the eye components, which is seven times two. Bless The magnitude of the product of magnitude of the J components, which is -14 times working. So this is equal to 14 -14, which is equal to zero, which means we got the daughter of the inspectors at zero, which means they are alternative victors. So I hope you understood the answer. Thank you.


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