5

Let Y be contractible. Let e be the constant map at yo € Y. Let & : [ _ Y be a map such that a(0) = 0(1) = yo. Must [a] be the same element of T1(Y, yo) a...

Question

Let Y be contractible. Let e be the constant map at yo € Y. Let & : [ _ Y be a map such that a(0) = 0(1) = yo. Must [a] be the same element of T1(Y, yo) as [e]? If so prove it, if not give a counterexample.

Let Y be contractible. Let e be the constant map at yo € Y. Let & : [ _ Y be a map such that a(0) = 0(1) = yo. Must [a] be the same element of T1(Y, yo) as [e]? If so prove it, if not give a counterexample.



Answers

Let f be an ordered field and x,y,z in F. Prove that if x<0 and y<z, then xy>xz.

Hello there. Okay, so for this exercise, we got this year a mapping. But first we need to define what is the meaning of this notation here on B you. And basically, this is the set of all. The ominous is, um, from VT. You can consider this just at all the maps, the linear markings, f from me to you. And that's enough. You don't need to consider anything else. So this is just a set of nothings, right? So now let's define this room mapping that takes elements from me to you. And basically, it makes all the vectors equals to the zero vector in the set. You. So that's why this got this sub index here you because this is the zero element on you. So what we need to prove is that this zero mapping actually corresponds to the zero element on the set of the Ami Murphy's, um, from the t you. So let's profess so let's consider any and nothing on the on the set of the Ami Murphy's um, from the to you, this is equivalent to say that f even mapping from me to you. Any money. Okay, then we know that for every V on the set. V we have that s plus and the zero mapping that we defined previously on this part of the veteran B It's going to be equal to f applied to be plus the zero mapping applied to be. By definition, this is going to be the zero back zero element on you. And we obtained that. This is echoes to F. B. So because f last oh, of B is equal to F B for every vector V on the set V then basically, we got the f lost. These year old zero mapping is equal to F, which is equal to the zero element plus f. So the zero mapping he is zero element on the R member, um, of b u under the addition right now, that was the first port. The second one say something really similar. So for the part, we we got that the negative of F on the set on the U is the mapping minus app that goes from the to you. Okay, so let's prove this. This is we're going to do in the same way as the zero factor. So let's consider any for everybody on V. We got that F plus minus one times f apply to this Victory V is equal to F minus f apply to the apply to this victory be and is equal to F B minus F of B. And this is just the zero factor on mhm zero vector of B, which is equal to zero. So what we got here is that first, we show that the zero mapping is equal to the zero element on the set of, um, the youth. So if we got that F minus, F is equal to the zero mapping, that means that this minus F is negative of F. So this implies that minus F is the negative in, um, the U.

Okay, so in this problem, we're given a linger transformation. Ron are in to our end. Okay, Now, the first question we need to answer is that whether whether verse exists, so you answer that question. The first note is that the minute information and B he noted as eight times X. So, um, by our definition, oh, onto so for any Why in rn there's always exists an axe from our end. Such that tee off acts was why so t up acts? It's just a few times acts just to hear, because here extra, it's the East available. I just inundated the available on the inside, but he actually is just a the mapping of acts under the transformation of tea. So it's just a T ax will be, Yes, because why so equivalent? Believe that means that the young to off this inner transformation implies the consistency off this in your system. So that means the system is consistent, sister. Okay, so by our bye convertible function convertible matrix theory, I I m t a is in vertical, so let's denote be to be the uber self hate. So we can find we can find a, um Lena Transformation. Asked which is staying first up t to be a inverse times axe and we substitute a inverse by. That will be be time, sex. So this is our, uh this is our universe off team. So ask. It's the inverse off t. So t is convertible, so that means t to universe exists. Okay, next question me to answer is whether the map in is on two. Um, to answer this question, we have to We still need to use a You see a bird convertible. Mac matrix. Incredible. Mission State here. Um, so we first consider the inverse up T, which is the axe. Now we follow the definition of round two. We just used a definition of onto to check whether this this new in your transformation will satisfy the definition of onto it does. Then we can say that is that is in your Central Asia, and he's on two. So take any why from our end. So what does there exist? Who, uh, that still exist and acts that makes as self eggs. He was white, your ex? And why are all victors Okay? So since the, um, the map enough x under the leaner transformation ass will be just We'll just be the p X. So this is saying as he acts, it was Why? So that is equivalent to say, um whether this linear system is consistent, whether the he's consistent. Well, the answer is yes, because we know he's in veritable. And since bees in veritable, we can apply over, he's a bit of a matrix here. Um, so that means that tells us B X equals why is consistent. So bye, I am t he actually was. Why is consistent? So that means there exists such an axe to make is to make a to make ass off acts. Because why? So that implies. Asked is too. So we're done. All right. Last question we need to answer is that whether the Lena transformation off in burst at his chambers, which is asked in this case, is 1 to 1. Well, this can't be. This can't be done. Just directly follow the convertible confirmations theorem because we know if B is in veritable and bye convertible matrix here on, we can at ease hards f just Do you know this part of my part? Off fingers? Incredible matrix the room. Um, we know that X two. Yeah, eggs is 1 to 1. And this is exactly our mapping ass are in to our end. So that means our as that means the transformation as he's 1 to 1. So we're done.

The transformation T maps The Vector X. Y. two. The product of the matrix times victor. The metrics is A. B C. D. The vector X. Y. That is the metrics ap city times to victor X. Y. Is the transformation of the picture X. Y. Through the map. T. Or its information. T. Show that in variant points other than the urgent exists if eight times the minus B times C equal A plus the minus one. So these information T goes from victor's of two components are too From Victor's of two components because the Metrics A, B, C&D. These two times or two x 2. And that multiplied by two times one victor give us a two times one picture. Anti of victor, X. Y. Is to find a symmetric ap C and D times a vector ex wife. That is. We multiply the metrics here times to give him victor and that result that result in victor is the image of the victor X. Y. Given. Now we talk about in variant points also called fixed points. These are points that has the property of having as the image through its information T equal to the same point. That is very important X. Y. In the plane as Birdie. That T of the point is it will the same point. There is the point is not modified when we apply the transformation T. It's called variant point. It is clear that the origin is an environment point. Because if we multiply the matrix times 00 victory. That is the origin. We get again to Syria zero victory. So 00. The origin is an in variant vector or any variant point. So we want to write the condition that need to be needs to be fulfilled in order to have in variant points other than these. Seriously a picture. So let's say that T of X. Y is equal to X. Y. If and only if T F X Y A B C D times X. Y. That's the same. Sorry, equal to X. Y. That is because of X. Y is defined this way. This is equivalent to let's multiply the metric stein's victor. We get a X plus B. Y. His first component, second component C. X plus dy. And that can be equal to X. Y. And equality of vectors means that correspondent components can be equal. So we got to have two equalities, E X plus B. Y equal X. And see eggs plus D. Y equals Y. And put in these terms to the left of the equations. We get a -1 Time six. That's when we put eggs to the left and take a factor eggs plus B. Y equals zero. And here with the same with Y we get C X Plus The -1. Y. people soup. Mhm. Okay. And this is the same as a matrix vector system. That is a linear system in X&Y. Which can be written a metrics form as a -1. Me first role of the metrics than C. The minus one times X Y equal serious. Ooh so this is a metric form of this linear system here. That is we want to find X. Y. Different from the version which satisfies his equations. So to have a non null solution to this equation. This linear system, it is uh equivalent to saying that the metrics is not invariable because if the metrics a no swan pc and even this one is variable multiplied by its inverse to the left and the only solution will be xy consider so. Okay, let's say that this way here linear system Amen, swan, E C&D -1 times X. Y equals serious. You as solutions different from 0000. Sorry if and only if The Metrics A Menace one, B. The U -1 is not either do has not been verse and that's the same. And this is equivalent. Two. The determinant of the matrix equals zero. Okay, equivalent to the fact that t determinant of these metrics be equal 20 And the determinant is not zero. The metrics is invariable. And as we said here, above the system, we have only the serious, serious solution. So we need this determinant to be equal to zero. That is the case in that case mattress has no reverse And this system get to have solutions infra France 00. There is determinant of the matrix. Amen is one. B C T -1 equals zero. Mhm. And we know this determinant is equal to the product of the elements on the main diagonal, A -1 times t minus one minus the product of the elements on the second, f minus B C. And that gotta be equal zero. And so we do this multiplication, hearing at a d minus a minus D plus one minus bc. He got zero. And so a d minus bc. That is left is to terms on the left of the equation equal. And the three terms go to the right of the equation. We get a plus the -1 which is just the condition said uh Name in the statement, there is this one here. So if this relationship between the interests of the metrics, A B c D defining the linear transformation is satisfied, then there will be in variant points of this transformation or to this transformation different from the origin. And that is because uh when we stayed the fact that it victor, x wire point X, Y. It's an environment point which is this equation here. This lead us to a linear system which will have non serious solutions. When the determinant of the matrix is equal to zero. That is when the metrics is invariable, it's not over. So the determinant is calculated and leads to this condition between abc and the Okay

Hello. Real question. Envisages when that F B and ordered field and X. So I said enough. Okay. It has also given that if X less than zero and why less than that then we need to prove that X. Y greater than access it. So let us get to hear that if access less than zero, this can be written as minus Act should be greater than zero. Okay, now here, if y is less than that so Zach minus Y should be greater than zero. Okay, no, these two have become positive quantities. Some multiplication of two positive quantities should be always positive, should always be positive. So we stretch it as minus X. Which is a positive quantity. Now into that minus Y. Which is again a positive wants to know should be positive. Let us open the bracket minus X. Z bless X. Y should be positive. Let us add except to both the sides will be having X way this is minus exceed all. It is minus except plus X. Y. And we are adding acceptable the sides greater than exit. So these two will become zero. So from here we are getting X. Y greater than X zet. So this is the thing we need to prove. Thank you.


Similar Solved Questions

3 answers
Chapter 8_ Section 2, Exercise 042 Condition gt requires numbers instead of "null" and "0" Consider the data with analysis shown in the following computer output:Dl SisuWhat is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means and differences in means?Round your answer for the pooled standard deviation t0 two decimal places.The pooled standard deviation isdegrees of freedom-
Chapter 8_ Section 2, Exercise 042 Condition gt requires numbers instead of "null" and "0" Consider the data with analysis shown in the following computer output: Dl Sisu What is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means a...
5 answers
Find each of the following: and 10. The angle 0 betweenIfa = <2,-3> and b = <-1,9.4its velocity the direction $ 308 E: Express speed of 14 milhr in IL.A ship is sailing at a vector in component form.14j and b = 2i 7i -are orthogonal.Show that vectors 12
find each of the following: and 10. The angle 0 between Ifa = <2,-3> and b = <-1, 9.4 its velocity the direction $ 308 E: Express speed of 14 milhr in IL.A ship is sailing at a vector in component form. 14j and b = 2i 7i - are orthogonal. Show that vectors 12...
5 answers
Solve for x:X m 1010X-2
Solve for x: X m 10 10 X-2...
5 answers
Omu~Uu Ie et uuuitithJn mal _ Tbt0LttDJr Eatk0 D 04221 {partua rabrabr4.0JaEAaMedLEcDSEInMei{en717b5MerhnnencJe we Sn[nHurJeluunaruJucetHeed Heln ?Ii na
Omu~Uu Ie et uuuitithJn mal _ Tbt0LttDJr Eatk0 D 04221 {partua rabrabr 4.0Ja EAaMed LEcDSEIn Mei {en717b5 Merhnnenc Je we Sn[n HurJeluunaru Jucet Heed Heln ? Ii na...
5 answers
8. (15 pts) Draw a complete arrow pushing mechanism(s) for the reaction below Please draw every arrow, every bond broken or formed, and the resulting regiochemistry and/or stereochemistry if there is any.H2.H OH HHH
8. (15 pts) Draw a complete arrow pushing mechanism(s) for the reaction below Please draw every arrow, every bond broken or formed, and the resulting regiochemistry and/or stereochemistry if there is any. H 2. H OH H H H...
5 answers
Look at the properties of the alkali metals summarized in Table $6.4$, and predict reasonable values for the melting point, boiling point, density, and atomic radius of francium.
Look at the properties of the alkali metals summarized in Table $6.4$, and predict reasonable values for the melting point, boiling point, density, and atomic radius of francium....
5 answers
Determine the driving-point impedance at the input terminals of the network shown in Fig. P12.5 as a function of $s.$
Determine the driving-point impedance at the input terminals of the network shown in Fig. P12.5 as a function of $s.$...
5 answers
5 f(r) dr = 2 ad $,5 gk(r) dx 6 , then 553f(.) + g(r) d
5 f(r) dr = 2 ad $,5 gk(r) dx 6 , then 553f(.) + g(r) d...
1 answers
Reasoning Is each expression in simplified form? Justify your answer. $$ \begin{array}{l}{\text { a. } 4 x y^{3}+5 x^{3} y} \\ {\text { b. }-(y-1)} \\ {\text { c. } 5 x^{2}+12 x y-3 y x}\end{array} $$
Reasoning Is each expression in simplified form? Justify your answer. $$ \begin{array}{l}{\text { a. } 4 x y^{3}+5 x^{3} y} \\ {\text { b. }-(y-1)} \\ {\text { c. } 5 x^{2}+12 x y-3 y x}\end{array} $$...
1 answers
In Exercises $5-10,$ tell whether or not $f(x)=\sin x$ is an identity. $$ f(x)=\frac{\sin ^{2} x+\cos ^{2} x}{\csc x} $$
In Exercises $5-10,$ tell whether or not $f(x)=\sin x$ is an identity. $$ f(x)=\frac{\sin ^{2} x+\cos ^{2} x}{\csc x} $$...
5 answers
Find sin 2x, coS 2x , and tan 2x if tanx = and x terminates In quadait [6, 150/sin 2xcos2xtan 2x
Find sin 2x, coS 2x , and tan 2x if tanx = and x terminates In quadait [6, 15 0/ sin 2x cos2x tan 2x...
5 answers
Find the interval of couvergence of the power series
Find the interval of couvergence of the power series...
5 answers
Problem 2.Now consider the case in Figure 2_ The geometry is the same as in Problem but a dielectricslab (blue) with dielectric constant # fills the bottom half of the capacitor On the left_(2)Q1Q2C14d2d 1
Problem 2. Now consider the case in Figure 2_ The geometry is the same as in Problem but a dielectric slab (blue) with dielectric constant # fills the bottom half of the capacitor On the left_ (2) Q1 Q2 C1 4d 2d 1...
5 answers
#Ntie: CKTCGee?atL 08 "3ircn Xar 841 4 AAr"iFFI % (Find ramp sire such thir crding %ckciy Erha rcquirrd platcau vrlony:) 1300*1Wm+?PE IGVRIa: {a+776"8763 IDitanc INJvodiztC Jrcj undertnc Lacity Erotilc_ LecthiioGkubic Nateiutimc:#umroond:
#Ntie: CKTCGee?atL 08 "3ircn Xar 841 4 AAr"iFFI % (Find ramp sire such thir crding %ckciy Erha rcquirrd platcau vrlony:) 1300*1Wm+?PE IGVRIa: {a+776"8763 IDitanc INJvodiztC Jrcj undertnc Lacity Erotilc_ LecthiioGkubic Nateiutimc: #umr oond:...
5 answers
What is the condensed electron configuration of the central metal cation for the following coordination compound? NHACr(OH)(CI)2][Ar]3d6[Ar]3d?[Ar]4s23d1[Ar]3d?
What is the condensed electron configuration of the central metal cation for the following coordination compound? NHACr(OH)(CI)2] [Ar]3d6 [Ar]3d? [Ar]4s23d1 [Ar]3d?...
5 answers
Y2 36x2 =1 4
y2 36 x2 =1 4...
5 answers
Letf(x,y) 2* ~ySolve the optimization problemmax f xys.tx+e-x =y
Let f(x,y) 2* ~y Solve the optimization problem max f xy s.t x+e-x =y...

-- 0.029956--