5

Compute the matrix of the linear transformation T Ps(R) 7 P(R) sending p(z) H plx + 2) , where Ps(R) is the space of polynomials of degree < 3, in the basis {23,...

Question

Compute the matrix of the linear transformation T Ps(R) 7 P(R) sending p(z) H plx + 2) , where Ps(R) is the space of polynomials of degree < 3, in the basis {23,z2 €, 1}. Use this to compute the eigenvalues and eigenvectors of this linear transforma- tion. Using what you know about graphs of polynomials, explain why this solution makes sense

Compute the matrix of the linear transformation T Ps(R) 7 P(R) sending p(z) H plx + 2) , where Ps(R) is the space of polynomials of degree < 3, in the basis {23,z2 €, 1}. Use this to compute the eigenvalues and eigenvectors of this linear transforma- tion. Using what you know about graphs of polynomials, explain why this solution makes sense



Answers

Let $T : \mathbb{P}_{2} \rightarrow \mathbb{P}_{3}$ be the transformation that maps a polynomial $\mathbf{p}(t)$ into the polynomial $(t+5) \mathbf{p}(t)$
a. Find the image of $\mathbf{p}(t)=2-t+t^{2}$
b. Show that $T$ is a linear transformation.
c. Find the matrix for $T$ relative to the bases $\left\{1, t, t^{2}\right\}$ and $\left\{1, t, t^{2}, t^{3}\right\} .$

Were given a transformation t from the set of polynomial of degree at most two from the set of polynomial of degree at most four, the maps pointing Will PT into the polynomial p T plus T squared PT. In part, they were asked to find the image of a certain polynomial. P f t equals two minus t plus t squared. We have that the image of this polynomial pft. Well, since it is a polynomial of degree at most two, it's going to be by definition, pft plus T squared pft and calculating. We have that This is the same as Tu minus T plus t squared plus t squared times two minus T plus T squared, which is the same is we have the constant to we have negative t we have t squared plus two T squared sets plus three t squared. Then we have minus T cubed and plus T to the fourth. So we get the polynomial to minus T plus three t squared minus t cubed plus T to the fourth, which is, in fact a polynomial of degree at most four in Part B were asked to show that this transformation is a linear transformation. So we're going toe. Let PFT and Q of T bee pollen or mules of degree at most to and see be any scaler. So will prove this in two steps. So first we have that t of PT plus q of t What we have that since both P and Q are polynomial of degree at most two, it follows that the degree of P plus Q is that most too, and therefore that the transformation is well defined for this point. Oh mule and gives us P f T plus Q of t plus T squared times, pft plus Q of T. This could be written as pft plus t squared pft After using the distributive property plus Q of T plus T squared Q of t. And we see these expressions in the brackets are simply t of p of tea plus t of q of t. So we've shown activity. Now we have that t of C times p of tea. Well, since he is a scaler, it follows that since the maximum degree of P is to maximum degree of C times, P. M T is still too. So this expression makes sense and is equal to see Time's pft plus T squared times see times p of tea and this could be written factoring out. See, we get C Times, pft plus T squared times Pft using the distributive property, which recognizes the Samos See Times t of PFT and therefore it follows that t is a linear transformation. In Part C were asked to find the Matrix for T relative to the bases, won t t squared and won t t squared cubed T to the fourth. So this time make it more explicit will have to. The basis be is going to be one t and t squared. The basis see is going to be won t he squared and cubed and then t to the fourth. So first, let's calculate the image of each of the basis vectors for from B So we have t of one is going to be one plus t squared times. One just simply t squared or one plus t squared. And so it follows that in terms of the basis factors, see, this is the column vector with entries one zero one 00 Likewise, we have that tea of tea is going to be t plus t squared times t which is t plus tty cute. And so we have that t f t respect to the basis. See, this is the column vector with entries 0101 zero t of t squared is equal to t squared plus t squared Times T squared, which is equal to t squared plus t to the fourth. And so we had the t f t squared with respect to See is the column vector with entries 00 101 Mhm. And so we have that The Matrix for T relative to these two bases. This is going to be the Matrix whose column vectors are t of one with respect to see t of tea with respect to see and t of t squared with respect to see which using our previous calculations. This is the Matrix, which has column vectors. 101 00 01010 and 00101

Problem questioning. So do you warn Be to Our Is big toe are serene. It's Amy. It was seen equally some staple one, perhaps and next. Sprayer and being. And that's slim. Something born X. It's weird and it's you don't so you know you one for one. It's equal to into, Oh, you want one, which is a pretty toe team Through or X plus one. Find one when she's April Brexit. Let's run it. Just exposed one man. So we put forward. Suddenly we're really lead to war. One. It went to It's squared, my sex, their industry toe X. All right, that's one. Me to you. One. Well, that's where x you not thanks Were they restaurant meals? Were there must lights there were trying to write these transportation using at the linguine, combining in or vector from scene So this can be right now one times one waas your plane. Plus, you find that script. This can be a little one times one Wellstone times Must your fun. It's spring when those can be written as your plans. One most two times. That's last three. Find it's It's play so accordingly. Using these confessions, you can say that need to be more off CNN day. It's secret to 10 wins here. I want to rent to you. So you know, for questions being or question being using the fear in our 6345 saying that Do you want on seeing a in tow ups? Evening you are. Oh, me. Hey, from in society training any? Thinks that aren't you? Want a Weren't one year deal. You won. What would you You know you want one in here? It's you. You You run this year on you. Number one You know you know you and you using the application for these two men tree Susan's We can not seem to want to be a It's important to zero euro foreign tour. You you question and seeing can you seems free for fun. And we're saying that into do you want off the s is different to you to work steaming well being. Oh, yeah. Remember of our yeezys? Well, you're in Really worry 12 And this is we're heading from here again. It's runnings. New ribbons in your own want to end you You know the application for a system nine races is able to seeps toe and since, But I want you to remember we were all the same the same so, uh, need to be worn. Oh, slur in one of those letters. Plus where once a squirrel. 2 to 1 of seven ones. That's us. We're in tow. It's a sport. Someone. Warren was lost for words. We're plus speaks. Thanks, Brian, must you? Jesus. What Must no seats X. That's where this Iran's nuisance suits to instance. What's your name? You, you know? Well, I need to do you work Seating. It's different of one times Two times Two times two minus warmly times U minus. Nearly times nice meal times. You your level, you know, thumbs. He usually sleeps. This is Mark in front of you, sir. Sense that runs nothing for you. Uh, it's you news about to you want It's not. You work too

Were given a transformation from, he said of polynomial of degree. At most two to our three i t. F p equals column Vector, whose entries are p of negative one p of zero in P of one. In part, they were asked to find the image under T of a certain polynomial. P f t equals five plus three t. So first of all noticed that P of tea, which is five plus three t, is in fact a polynomial of degree less than a record two. So this doesn't make sense. Thanks. Well, we have that. The image is going to be a column vector whose components are P of negative one p of zero and P of one. Then we see after plugging in these values, he of negative one is five minus three, which is to P of zero is five plus zero, which is five and p of one is five plus three, which is eight. So we get to column Vector 258 in Part B. We were asked to show that this transformation is in fact, a linear transformation, so we'll let P of tea and Q of t the polynomial of degree at most two and well, let's see be any scaler. Then we have that image of T plus or P plus. Q. Well, we have the both P and cure applying the moves of degree at most two. Therefore, the degree of P plus Q is also at most two. So this makes sense, and this is going to be the column. Vector, whose entries are polynomial P plus Q. Evaluated at negative one polynomial p plus Q. Evaluated at zero and P plus Q. Evaluated at one. Now we know that the addition of Polynomial is that a point is the same as Plano. Meals at a point added together. So this is the same as P of negative one plus Q of negative one P of zero plus Q of zero and then p of one plus Q of one. This can, of course, be written as thesis, um, of column vectors. He is negative. One p of zero p of one, plus the column. Vector que of negative one que of zero q of one we see it. This is the same as T of P plus T F. Q. Now we have the t of c times p well, because P is a polynomial of degree at most two, it follows that the scaler see Times P is also a polynomial of degree at most two. So this makes sense, and we have by definition, this is the column. Vector components. See Time's p evaluated at negative one. See Time's P evaluated at zero and c times p evaluated at one. And we have that because see is your scaler. You could be right. This as see Time's P evaluated at negative one. I think this is called quasi associative Lee. Maybe then we have C Times p of zero and C times p of one and can factor at a C from this column vector to give us simply c Times the column Vector. You have negative one p of zero p of one, which is the same as C times T of Pete is what we wanted to show and therefore follows. The tea is in fact, a linear transformation. Finally, in part, C were asked to find the Matrix for T relative to the basis won t t squared for p two in the standard basis for our three. So we have the basis p two won t he squared. So called be the basis See for our three, which is simply e one e two and e three. So to find this matrix, I want to compute each of the basis vectors of P two in terms off the basis vectors of our three. So we have that t of one. This is going to be column vector whose entries are mhm one evaluated at one just simply one verse. Everyone value at negative one, which is 11 evaluated at zero, which is also one on one of value. That positive one, which is also one. And this is the same. Has column Vector 100 plus the column. Vector 010 plus the column. Vector +001 which is the same has E one plus e two plus e three. Likewise, we have the t of tea. This is going to the column vector with entries. Well, t of negative one is negative. One T f 00 in the city of one is one, and this is equal to the sum of column victors. We have negative 100 plus 001 and this is the same as negative e one plus e three. Finally, we have that tea of T squared. This is going to be the column vector with entries. He squared value that negative one, which is simply one, then zero, then a positive one. And this is the same as the sum of column vectors 100 and column Vector 001 which is the same as E one plus e three. So going back, we have that the image of one with respect to basis. See, this is going to be the column vector with entries 11 and one coefficients of you. Only 23. We have that the image of tea with respect to the basis. See, this is the column vector of entries negative 10 one. And we have that. The image of t squared with respect to see is e column vector with entries 101 And so we have that The Matrix 40 relative to these bases is going to be the Matrix with column vectors the image of one with respect to see image of tea with respect to see and the image of t squared with respect to see which we calculated is the Matrix. With column vectors 111 negative 101 and 101

Okay for problem Problem 10. We have the leaner, faster mission maps from P three to Arthur are poor. Such that t of pulling nominal is equal to by taking different input to the Polo Meo. Just pee of connective three first p A connected one and p of one and p up three. Hey, So the first thing we need to show that is that, um to show this transformation is linear. Okay, so do that. My transformation. We can first check tee up. People ask you where panic you are all clean on yourself. Order to be so then, by our assumption, we can have a p rescue while we take the include to be elected three. And he ask you with the input elective born people ask you with include one and people ask you with your food three. Now, because we are adding these two point nominees so we can we can calculate the number Sweet that with this input on dad them together just just by by the rules, our tradition. So that's he, uh, 93. I trust you collected three and PR. Negative one. Uh, you, uh, negative one and p of one less queue of one. And here, three askew on three. So by now we can separate this, uh, this vector with by a vector off Minami O P and the vector polynomial Q. So it turns out to be thio he that's t l Q. So that's the first part here. So the second part is considered skater Linda times people. So again, by our assumption, we still have Lunda cons. P off negative three Lambda SPF connective one, huh? Times p of one and Linda Times p of three. Excuse me. So we can take out the skater from our rector's. So we have loved a in front of our specter and P uh, negative three. He, uh, connected one he won and ps three. So that turns out to be Lunda time. So? So we're done for the first part because I our this relation and second relation. Then we can conclude that he's a renewed transformation. Now, Part B. We need to find the matrix for tea relative to the basis won t t square t t cubed for p three and standard basis are for Excuse me. So now let's, um defined business P start this is B to be given basis in our assumption, which is one two he squared and t Q. So we first calculate, uh, actually the tea with the input up won t t square and teeth cute separately. So first we have to be one which gives our vector all once and then we're plugging t which gives Dr Inactive three negative one, one and three. Then we're plugging our t squared. This gives Excuse me. Um 911 night, and at last we plugging our thank you. So we had a vector collective 27 negative. 11 and 27. So our metrics will be this band of these four vectors. So this will be he of this is B should be one collective three nine Next 27 one defective. 111 1111 and 139 27. And that's it.


Similar Solved Questions

3 answers
Let p and qare two statements, then (~p € 9) - (~p ^ -q) is logically equivalen'a) p ^ q b) 7p ^ -q c) p V q d) 7p V -q796980 | @ =ak icin buraya yazin
Let p and qare two statements, then (~p € 9) - (~p ^ -q) is logically equivalen' a) p ^ q b) 7p ^ -q c) p V q d) 7p V -q 79698 0 | @ = ak icin buraya yazin...
5 answers
Which of the following will cut at the fewest sites in large chromosome 10 million base pairs long ssume its quence is random)0 A restriction enzyme withbase-pair recognition equence0 A restriction enzyme withbase-pair recognition sequence 0 A restriction enzyme with an base-pair recognition equenceCRISPR/Cas9 guided by ~ 21 base pairs of base pairing to guide RNAAll would cut equally frequently
Which of the following will cut at the fewest sites in large chromosome 10 million base pairs long ssume its quence is random) 0 A restriction enzyme with base-pair recognition equence 0 A restriction enzyme with base-pair recognition sequence 0 A restriction enzyme with an base-pair recognition equ...
5 answers
Find the position of bcdy moving on coordlnalc Iine at timc for an aczcleration-2e initial velozity v(O) = 10, and inlbal position #/0} = 5.The position 0f the body at tima t is given by $ =
Find the position of bcdy moving on coordlnalc Iine at timc for an aczcleration -2e initial velozity v(O) = 10, and inlbal position #/0} = 5. The position 0f the body at tima t is given by $ =...
5 answers
Be able to identify the molecular formula based off of structure: Example molecule: CHz CH;CH3HO-Capsaicin
Be able to identify the molecular formula based off of structure: Example molecule: CHz CH; CH3 HO- Capsaicin...
5 answers
Bnegative charged particle Is placed in magnetic fiold. The force acting on the charge Is In what direction? Northb. Out of the page WestSouthThore Ig no Iorce acling on the particle East '0 Into the page.
B negative charged particle Is placed in magnetic fiold. The force acting on the charge Is In what direction? North b. Out of the page West South Thore Ig no Iorce acling on the particle East '0 Into the page....
1 answers
DoLdTo5Quostiou 6: (3 Pointo) Lut p b prite iuteger. Suppose Lht 4 is 4 intcger guch tlut p divides Show that p divldc u
Do Ld To 5 Quostiou 6: (3 Pointo) Lut p b prite iuteger. Suppose Lht 4 is 4 intcger guch tlut p divides Show that p divldc u...
1 answers
Evaluate by using frst substitution and then partial fractions if necessary. $\int \frac{x d x}{x^{4}+1}$
Evaluate by using frst substitution and then partial fractions if necessary. $\int \frac{x d x}{x^{4}+1}$...
5 answers
The percentage of variation in the valuesof y explained by the least squares regression lineisA) the y−intercept of the regression line.B) the coefficient of determination.C) ρ.D) the correlation coefficient.E) the slope of the regression line.
The percentage of variation in the values of y explained by the least squares regression line is A) the y−intercept of the regression line. B) the coefficient of determination. C) ρ. D) the correlation coefficient. E) the slope of the regression line....
5 answers
OulatiowDir 4m0 di7dp1605 d @urdnn#K60nJ0c4- JOtm udq"} 0nCax=JOc Find thc mputude oftkWLc 0JVLc SEc GEcWEc
oulatiow Dir 4m0 di7dp1605 d @urdnn#K60n J0c4- JOtm udq"} 0nCax=JOc Find thc mputude oftk WLc 0JVLc SEc GEc WEc...
5 answers
[0/1 Points]DETAILSPREVIOUS ANSWERSSCALC8 3.9.037_MY NOTESASK YOUR TEACHERPRACTICE ANOTHERFind fsin(0) cos(0) , K0)f '(0) = sin ( 0) cos( 0) 50 + 3((0)=Need Help?ceeVadng Saved Work Revert aeeResddnga
[0/1 Points] DETAILS PREVIOUS ANSWERS SCALC8 3.9.037_ MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find f sin(0) cos(0) , K0) f '(0) = sin ( 0) cos( 0) 50 + 3 ((0)= Need Help? cee Vadng Saved Work Revert aeeResddnga...
5 answers
Please give a description for why you picked thisanswer Which statement is INCORRECT?Aerobically, oxidative decarboxylation of pyruvate formsacetate that enters the citric acid cycle.In anaerobic muscle, pyruvate is converted to lactate.In yeast growing anaerobically, pyruvate is converted toethanol.Reduction of pyruvate to lactate regenerates a cofactoressential for glycolysis.Under anaerobic conditions pyruvate does not form becauseglycolysis does not occur.
Please give a description for why you picked this answer Which statement is INCORRECT? Aerobically, oxidative decarboxylation of pyruvate forms acetate that enters the citric acid cycle. In anaerobic muscle, pyruvate is converted to lactate. In yeast growing anaerobically, pyruvate is converted to ...
5 answers
Explain how Gibberellic acid affects the growth of a plant.Provide articles as evidence.
explain how Gibberellic acid affects the growth of a plant. Provide articles as evidence....
5 answers
Given that csc(x)=7/6, what is csc(−x)?
Given that csc(x)=7/6, what is csc(−x)?...
5 answers
Please show all work and explain please, thanks.
Please show all work and explain please, thanks....
5 answers
Mass of reactant_8Moles of NaHCOz reacted_2 molesMass of product NaCl:8Moles of NaCl produced_molesExperimental Mole Ratio NaCl to NaHCO3-Theoretical mole ratioPercentage error of experimental mole ratio_
Mass of reactant_ 8 Moles of NaHCOz reacted_ 2 moles Mass of product NaCl: 8 Moles of NaCl produced_ moles Experimental Mole Ratio NaCl to NaHCO3- Theoretical mole ratio Percentage error of experimental mole ratio_...
5 answers
Use the given information to find the exact value of the trigonometric function cos sec 0 = 4, 0 lies in quadrant /Ve = 2715 OAV8+2+715 0 B.5o OD. 5Cllck to select your answor:
Use the given information to find the exact value of the trigonometric function cos sec 0 = 4, 0 lies in quadrant / Ve = 2715 OA V8+2+715 0 B. 5o OD. 5 Cllck to select your answor:...
5 answers
~12 POINTSSPRECALC7 8.3.040_Write the complex number in polar form with argument 0 between 0 and 21 .Need Help?Rsad IoJalkto Wuler
~12 POINTS SPRECALC7 8.3.040_ Write the complex number in polar form with argument 0 between 0 and 21 . Need Help? Rsad Io Jalkto Wuler...

-- 0.020589--