let-t-mathbbrn-rightarrow-mathbbrn-be-an-invertible-linear-transformation-explain-why-t-is-both-one-

Name: let t mathbbrn rightarrow mathbbrn be an invertible linear transformation explain why t is both one
Uploaded: 2020-12-07
Duration: 6 min 2 s
Channel: Numerade
Description: Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be an invertible linear transformation. Explain why $T$ is both one-to-one and onto $\mathbb{R}^{n} .$ Use equations (1) and $(2) .$ Then give a second explanation using one or more theorems.

Question

Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be an invertible linear transformation. Explain why $T$ is both one-to-one and onto $\mathbb{R}^{n} .$ Use equations (1) and $(2) .$ Then give a second explanation using one or more theorems.

Ahmad Reda · Accepted Answer

in problem 35 If we have tea and in veritable linear transformation, it&#x27;s an in vertebral. Then you&#x27;re transformation. We want to explain why he is both Oneto one on gone to our end using equations one and two. Because he is a linear, veritable linear transformation, it means it&#x27;s a transformation. For example, this system X equals six we want when we want to blow it by t we&#x27;re t is the inverse off T is the inverse off the which means we have the equation. One is given here and the second equation is 80 x equals x no for the first part of the problem for one toe one proof we want to prove that the X equals zero has the only reveal solution. We can assume X as any victor Such is that Z for example? Then we have TV equals zero demand by by V equals zero and boy applying the embers off t. We must apply both sides by a from left. Then we have a T V equals a multiplied boy zero that she gives zero. We can see that we have a TV, the left hand side and the left hand side is very similar to the equation, too. Where e t multiplied by a victor gives the same victor. Then we have the left hand side equals TV from Equation two and from the right hand side, it equals zero. This means this equation has only the trivial solution. We have proven that T V equals zero because we will be only equals zero. We will not be in a victor. V will be all those zero, which means the X has only trivial solution. And this proves in the first part of the problem for the second birth on to are in we want to prove that t multiplied by X equals B as at least one nontrivial solution. As at least one solution, you can prove this the same way as 1 to 1. Bye, I think I mean, you have t want to blow it by any victor. For example, W example w equals B. We want to prove that be has a solution. Is is not always be zero solution like this. We use equation too and multiply both sides from left by the inverse of T, which is a then we have 80 w equals a B and from equation, too. We have 80 multiply by Any Victor&#x27;s gives the victor, then w equals the metrics. A multiply it by affected b will be is the non zero victor, which makes the right hand side, is down zero and has a solution. Which means that this equation gives a solution X equals w. And this proves the second board of the problem. And we have from fear, um, it we can prove that if mhm is in vertebral metrics, then it&#x27;s a transformation. For example, T is 1 to 1 1 to 1 and it maps are in home to our in from the in vertebral metrics theory, and this is the final answer of all broken.

Runpeng Li · Answer

We need to show that, as is a linear transformation and were given the condition that, um, t and s satisfy the credibility creations and tease and enough inspiration. Okay. How? Just write down our goal here. You near transformation. Okay. All right. He ordered to do that. We take arbitrary, take any you and B from our in. So we pluck. Are you gonna be to transferred to the ingenious information as we will have X? And why accident. Why, Okay, now we consider t off acts. Now, remember, acts can be represented as ass off you. So plugging as self you now, since ass is it, it&#x27;s the number. As an S and T said it&#x27;s by the embers in credibility questions. So this is this keeps us directly back to you. So similarly, we have, um, t apply is factor VII. Okay, so now we consider ask you plus. Yeah. So since you plus V can be represented as a transformation of x and y under T, So that will be t ext plus t y. Now remember the inner transformation by our assumption, the transformation transformation T is a leaner transformation so we can apply the property. Help me near says Formacion here at his t x s t y t X plus y okay. And by our convertibility equations, these keeps us express What? And recall that X is s you And why is SP so that ask you Class sp and our first condition It said his buy. So the next condition we need to check it&#x27;s whether s off the escape, their off vector under the interview and then you&#x27;re under the transpiration Information off ass will be the same as the skater times times the transformation off s so that is directly to show that c o s u because s self see you as I see you, that&#x27;s it. And to do that first consider considered TLC X because he&#x27;s a linear transformation. So it&#x27;s just a tee off a sea of t x c times t x and T acts by our previous by our previous reasoning here, T X will be you. It&#x27;s C times. You okay? So s off. See, Time&#x27;s view can be written as as soft tee time. See x right, which is a substitute c times you by. I don&#x27;t trust for Michel t so That&#x27;s, uh, tee times tfc times eight times X. Now we can we can apply the convertibility equations, so this gives us see X. Now, remember, ax by over by this equation here, acts can be represented as ask you so at a same as C times Ask you so as I see you is C times as here. So we&#x27;re done. That means ass, he said.

Angelo Rendina · Answer

say we have the system A X equals B and disease Stoma solutions for every vector be in RN Now, if a he&#x27;s not in veritable, then we reduce the system metrics, eh? Me to something like upper triangle medics here and then on entire roll off zero at the end and appear this whatever. And then here we have the transformer coefficients so easy A is not in veritable. But that means that for some be hence be teal that the system cannot have solution against assumption that the system had solutions for every vector B. And of course, the problem is that we&#x27;re think we&#x27;re assuming that B is not in veritable so for the system to have solution always, the medics say, has to be in vertebral.

Rakvi . · Answer

recalculating words in this problem that is, or two by two matrix whose entries are as follows. This is minus for minus one. And so So, as I said, we want to find inwards, Sophie. So let us first get played your German and off you literally. Nando is minus four times to which is minus it minus two times minus one, which is minus two and minus eight. Plus two is minus six. Yeah, and now in worse is just one upon determinant trophy times a matrix which is obtained from co factors off the original matrix. So in particular for two by two matrices You have that you just entertains thes two entries and take negative all thes two other entries, so you&#x27;ll get to minus four and here you will get one and minus. Ah, so this becomes two by minus six. Store divided by minus six is minus one, divided by three and minus one, divided by six minus two divided by minus six is just 1/3 and minus one, divided by minus six. Who just toward so that&#x27;s your in worse. And you can quickly check that if you multiply a everyday in Worse, we will get I didn&#x27;t

Rakvi . · Answer

We just want to find a and was in this problem. That is easy. Quartile. 1123 No determinant of is is three times one minus two times one, which is three minus two, which is one. Okay, so to obtain inwards, you just take the matrix obtained from co factors and divide by determinant, which is one in this case. We just take the matrix obtained from the co factors. So you will get 31 minus one and mines. So that&#x27;s your angels.

Michael Jacobsen · Answer

in this exercise were given a set of linear dependent vectors in R N Let&#x27;s start out by writing just what it means for this set of vectors to be linearly dependent, it means is that what this means is there exists scale. Er&#x27;s x one. Let&#x27;s go with C one rather see to and C three, which are not all zero such that the following linear combination is satisfied. Where we take C one times V one plus C two times V two plus C three times V three and this will be equal to the zero Vector where we said not all of these vectors air Zero. And that means that this vector equation has a nontrivial solution. Let&#x27;s inject a new element into this situation. Let&#x27;s define T, which will map from the vector space are in into RM. So that t is linear here. The most important statement about T is that this transformation is linear. So now we wonder about that are set of vectors that could be the images of tea that would be specifically TV one TV too, and TV three we know that the set of vectors v one v two v three is already linearly dependent. So the question becomes, What about this set? Should it be linearly, independent, linearly dependent? Or do we have not enough information to tell? Well, one way to determine the outcome here, using our transformation T is to start back up at this equation here. Let&#x27;s apply the transformation t on the left hand side. So that&#x27;s tea of C one times V one plus C two times V two plus C three times V three. Then on the right hand side will use the fact that since T is linear t of the zero Vector is still the zero vector so we can write in the zero back there here. Well, this equation so far, it&#x27;s giving us closer to determining if this set is linear, independent or dependent. But we haven&#x27;t gone further enough. Let&#x27;s begin using more properties of linearity. Linearity allows us to write this left hand side here in the form of C one times T a. V one plus C two times TV, too, plus C three times t at V three and this is set equal to zero vector. Sylvie in your itty allows us to break apart the vector addition as well as the scaler multiplication so that we get to this new equation. Now let&#x27;s recall what we said about C one, C two and C three. We said that they are not all zero, and here they are in this equation. So that means that this vector equation has a non trivial solution. And it&#x27;s also a homogeneous vector equation because the right hand side is equal to the zero vector. Now see what it means for a set to be linearly dependent. This set is linearly dependent now because we have a nontrivial solution to this equation, and so we can now state our conclusion TV one TV, too, and TV three as a set is linearly dependent. So the punchline to this example is that linearly dependent sets are mapped into linearly. Independent sets provided our transformation t is linear now, sadly, we might want to extend this. What if the set was linearly independent? Well, tragically linearly. Independent sets could still be mapped to a linearly dependent sets. So the statement here that this set is linearly dependent on Lee works for dependence

Problem

Let $T$ be a linear transformation that maps $\ma…

Problem 35 Easy Difficulty

Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be an invertible linear transformation. Explain why $T$ is both one-to-one and onto $\mathbb{R}^{n} .$ Use equations (1) and $(2) .$ Then give a second explanation using one or more theorems.

Answer

one to one
onto
onto

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Heather Z.

Alayna H.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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one to oneontoonto

Linear Algebra and Its Applications

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Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Heather Z.

Alayna H.

Michael J.

one to one
onto
onto

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications