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Each of the following are transformations from Rn to Rm For each; determine: (i) the values of n and m (i.e., is f a function from J2 3 R5 , etc:) , (ii) whether th...

Question

Each of the following are transformations from Rn to Rm For each; determine: (i) the values of n and m (i.e., is f a function from J2 3 R5 , etc:) , (ii) whether the transformation is linear, and (iii) if the transformation is one-to-one, onto, O neither. T1 + T2 81 (a) f %1 82 T2 0T1 82 83T1 282 + T3 T2 + T3 T1 +5(b) fx1 T2 83 T4T1 + *3 T2 + T4(c) f

Each of the following are transformations from Rn to Rm For each; determine: (i) the values of n and m (i.e., is f a function from J2 3 R5 , etc:) , (ii) whether the transformation is linear, and (iii) if the transformation is one-to-one, onto, O neither. T1 + T2 81 (a) f %1 82 T2 0 T1 82 83 T1 282 + T3 T2 + T3 T1 +5 (b) f x1 T2 83 T4 T1 + *3 T2 + T4 (c) f



Answers

In the following exercises, the transformations $T : S \rightarrow R$ are one-to-one. Find their related inverse transformations $T^{-1} : R \rightarrow S$
$x=4 u, y=5 v, \quad$ where $S=R=\mathrm{R}^{2}$

It will come back them Rate one question 372 of chapter five, section seven and the cat the suit textbook. We're given the linear want one transformation X equals you plus the plus w And why equals why it was three v and Z equals to w So can we find use a function of x, y and Z? We find these a bunch of ex Wednesay confined the W's projects of wine seasons thes air one toe, one transformations. The inverse is worse exist. Well, I'm going to start here. What if I did find both sides by three? Then again, V equals y over three. Okay. And what if I did? By both sides here but to and we get that W Izzy over too. So the view's a function of that 20 hz. And these a bunch of ex wife teaches wise This form s so now what? Well, if x is you, plus V, but V is why over three and plus w and W Z over too. Can I get you long? Yes. Attract both sides by Wyman over three and Z over to. So you becomes X minus y or three minus Z over two. And so you've got interest functions. We're good. All right. The radio. That was helpful

Okay, Memory will come back for the question. 3 70 off jet 56 or seven in the captain's three textbook. Given the transformation, linear transformation, X equals eat of the children, you plus V and why equals eight of the humanist? All right, so since these this oneto one transformation that's given, we don't do that in this particular question. Can you now get you as a function of X and Y? And he's a function of X and y What if I take the log natural level both sides. So we get the Elena vets is equal to you, plus V, and you get the Ellen of wine his you minuses. So I'm trying to separate the using the visa, get use a function of X and Y and these much of it. Look at this. What if I add these two expressions together the V's cancer so I can focus on you? All right, let's do that. So ln of X plus ln of why is, um it's to you is to you Plus you. So three years. So what's you gonna be? It's gonna be you's gonna be 1/3 of L a max facility. Why But isn't that the ln of x times where you could write Ellen of experts? Let's see here. Yeah, that's something. Right? Let's clean it up a bit. 1/3 the ln of experts. See, Ellen, why is the land of Mexico? It's a lot of product into some boys. Likewise. What, Faison? Strength ese, too. Let's see what is attractive. So you get the Ellen of X minus the Ellen of one. The Ellen of Vets minus the L in a war. This minus this. What do you get? Well, you get you okay. So to you once you with you and the minus and they give these a B plus one is two. Yeah. What is that? Well, we know what you is. It's the Eleanor, that swine. But this. What about this? What if it's a crime to Ellen wise and cancel the use? That's what I want. I don't want to get rid of you know, I got rid of the V. Now let's give her the sorry. Our districts let students, huh? What if you take away two wise, uh, good to Ellen? Wants what Second? Well, particular Timex to you. Close. We've minus two Ellen White's Which to you minus two. Big than you have to you. Minus two, you cancels. And three minus a negative to the is three plus two. These three a And, um, So then what do you get? Well, then you get the dean is 1/3 of X minus two. Ln Why? But tooling Weisz, Eleanor y squared and Eleanor ex went selling. Why squared is Elena Victor? How about that? Because the Elena potion is the difference of the lens and the Ellen of power. A limb of y squared is tthe e Excellent, right? So they helped those helpful YouTube star numerator and gullible.

This is question two from section one point one, vertical and horizontal translations of functions. And we are given the graph F of X. And so I have put the graphs off to the right hand side here and you can notice that I've placed at different places on the graph so that I have room to do my translations. So we are given an equation for fx. So we are going to look at the translation that G fx creates and we can see here that H is equal to zero and K is equal to three. So that means that there's no vertical translation. But the graph is going to move up by one. One of the probably more efficient ways of doing it. If you have a graph that's easy to do this is just take each point and move it up by 3123 And then be goes that's a prime and be goes up 123 that's be prime and see prime. And the prime getting and we can do a general connection here. So that is our F of G of X. And we need our points. So that's a prime is negative four. And one be prime is negative three and four. And see prime is negative one and four. And deep prime is one and two, and e. Prime is two and two. I would notice that are Y value is what changes on this one, and we add three to each Y value. So our mapping would have been X. Why next last three? So we could figure out the new points each way through and then graph it later. If we wanted to I'm going to stick to this. So H of X. We know that H is equal to two and K is equal to zero. So now we have a horizontal shift and it is going to the right to so I'll take my points again and moves over by 212 That's a problem with the green be moves over by two. That's that's be prime. Siemens over by two. Thank you. Mhm. And then we have DNA and so we can connect the dots again and you can see it's the same shape that's shifted over and copy out the values again. We've got a prime is equal to I could have to and negative chimp B think of one park see 11 D three negative one and E. Is for and make it. Mhm. Yeah. And again if you would look at if you have the original points you just noticed we were adding to to each X. Value. So my next one is sfx is equal to f of X plus four plus four, tells me that H is equal to negative four and K. Is equal to zero. So all my ex values are going to go to the left by negative four by four. So A goes over four B. See and we can connect the dots again. A prime negative eight negative to be prime negative seven point. Mhm. Mhm. Mhm. Yeah deep prime negative three negative one and eat prime negative two negative one. And again it's the X values that changed. Lastly we have a shift that's going down H. Is equal to zero. K. Is equal to to negative two. And so it means we're going down by two. So again countdown. Mhm. April feet fried. See prime the prime if yeah you got the general shape again and so our points the prime is negative three negative one. See Crime is negative one negative one. The prime is 93 1. D. Prime is one negative three. And e prime is too negative three.

You're right. Welcome back. We're in question 3 68 in chapter 56 and seven of the Captain's suite Textbook on that were given, um, the 1 to 1 transformations from artists is X equals for you. And why equals five V. All right. And so I don't think so. All right, so they want it. Since it is a 1 to 1 transformation, that should be an inference, right? S. So, in other words, can you find you as a function off axel? Divide Will says Ray for. And you get that you is exit before. And can you find a visa? Fortune White? What the Bible says about five and the equals Y or five. So the inverse exists in that, isn't there? All right, that's it. Through that question. Wait. That was helpful.


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