consider-the-polynomials-mathbfp_1t1t-mathbfp_2t1-t-and-mathbfp_3t2-for-all-t-by-inspection-write-a-

Name: consider the polynomials mathbfp_1t1t mathbfp_2t1 t and mathbfp_3t2 for all t by inspection write a
Uploaded: 2019-08-11
Duration: 1 min 13 s
Channel: Numerade
Description: Consider the polynomials $\mathbf{p}_{1}(t)=1+t, \mathbf{p}_{2}(t)=1-t,$ and $\mathbf{p}_{3}(t)=2$ (for all $t ) .$ By inspection, write a linear dependence relation among $\mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3} .$ Then find a basis for $\operatorname{Span}\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$

Question

Consider the polynomials $\mathbf{p}_{1}(t)=1+t, \mathbf{p}_{2}(t)=1-t,$ and $\mathbf{p}_{3}(t)=2$ (for all $t ) .$ By inspection, write a linear dependence relation among $\mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3} .$ Then find a basis for $\operatorname{Span}\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$

Doruk Isik · Accepted Answer

in this problem. We&#x27;re given a set up for real, and we&#x27;re basis. Uh, we&#x27;re in space space formed by this Fulham&#x27;s. So using this, people almost weakened. Sugar Petri is actually P one plus p too soapy. Three can be written as a linear combination off other two colonials. So spending said, hero, this face spent by old people in those you want me to be three the legal to the space spend by only using only Poland off you and Peter. Since Petri can be written as a communist off, you wanna pee too. And by looking at the person a second Hold on till we see that people and p two are we nearly dependent. So since they spend in space and since thoroughly new dependent on the basis for this system will be if you are and if you want and eat

Doruk Isik · Answer

Yeah, in this trouble were given a set up. So we&#x27;ll start solution by writing the organ directors for each polynomial. So for people in the quarter, Inspector Ho. But like this 101 said, the 1st 1 is the question. But these year old 2nd 1 is to one of you two. So using this Pete, too apartment that too perfect. You could be rich. Zero. What being? And for P three, that could be, um, that would be one one negative three. So then the augmented matrix phoned my bees. Corn factors would be 101 01 negative three and 11 Negative three. Now let&#x27;s write this over mental midgets in traditional echo form. And that is he could do warn. 00010001 is you can see. We have three columns and each column has a few bits elements. So three columns, three events elements at each column. So it means that these vectors Spector&#x27;s spend our fee. It means that the given Colonials no meals spin he two. Now in part, we were given 1/4 collector for a loan Jew, and that is next to 112 And whereas to find this bologna to using this So we know that cube of native one times people plus one p two plus thio three So Dunn will be negative one times ones You&#x27;re one plus one times zero wounding three waas Two times One wonder three And we find this to be one the next 10. And using this, uh, using the information that these numbers represent corruption tease. We can ride the roller milk you is one plus B t minus 10 t spring.

Runpeng Li · Answer

work, even Pelino meal overseas. One, then one minus t then to minus 40 plus T square. The last one is six minus 18 T. That&#x27;s nine. T squared us. Sorry. Minus cube. Okay, so in terms of the Matrix we have of four by four matrix. So it is the first efforts column represented a constant term. Off off. Pretty nami second column represent our one and so on and so forth. We have off one are off to and our off three. So for the person in question, we only have one and other things will be zero. The second equation. We have one constant term and, um, negative one for that t and also right. There are two zeros and the third column. We have two for constant term and negative war or tea, one for t squared. And for the last column, we have six accounts in term negative 18 for teeth and nine for t squared. And next one or T. Cute. Okay, so it&#x27;s no hard to observe that they are four people. Two columns. So this four is exactly, um excuse me. So these four people columns, uh, exactly equal to the number off. Go to the number. Oh, pretty no meals given complete onions given in our meals. Nam Joo&#x27;s. So that means these columns for columns are columns. All right, the nearly independent. Okay, so the answer is yes. This is four. Really? Nah. Mills will spend the whole p three. So? So these four? Well, you know, meals lean on mules. It&#x27;s been the three, so we&#x27;re

Angelo Rendina · Answer

we have your focus now. Basis zero p one Batou, given by the polynomial is won T and T Square Minister. We know these from example Five. So I mean, let me right zero bar for the vector obtained by computing P zero in minus two. Imara swan in zero in one and into So this is just the vector off ones and then be one bar is minus two, minus 10 and two and finally P to bar. He&#x27;s two minus one minus two minus one into. Now we write these vectors because we&#x27;re called the dinner plucked between fully normals. So in this case, for instance, PCR against zero he&#x27;s given by p zero bar don&#x27;t p zero bar. Sonny&#x27;s get is five, and similarly, we compute the one against the one to be 10 and p two against me, too, to be 14. We complete these numbers now because they&#x27;re going to use them now. We&#x27;ll soon when we&#x27;re doing the Grand Street process. So let&#x27;s go see there The Perimeter Cube, which is given by D Cubed. Let&#x27;s right, Q bar, of course, is minus eight minus 101 and eight and we want to apply the garnish meat process to Q to obtain well obtain up vector riches or for colonel to both be zero p one MP too. So now the computation is simplified by the fact that the vectors p zero be wanting me to our world vectors. The polynomial sze p zero we won&#x27;t be too are already or for Colonel. So all we need to do is to compute the projection off cue unwto p zero p one MP to Palestine would be zero, which is Q against zero divided by P 00 And of course, we recall that these is Q bar dot p zero bar divided by five the fire we computed previously and then P zeros of the constant function one and now you can compute a cube are 0.0 bars zero So this whole projection is the no vector. Then we compute the projection on Topi one off you seeming our lady this cure against the one divided by the one against the one you compute his number. It is the result to be 34 divided by 10 and then be wanted as the constant Well, the function t That&#x27;s what these 3.4 t and finally the projection onto P to off you He&#x27;s again Q against me, too divided by P two against me. To that multiplies the polynomial P too. And again you against see two turns out to be just zero. So this whole number is really bothered by 14. That multiplies the vector with very normal D squared minus two and he&#x27;s just zero. And therefore we can put the three to make you minus all these projections. And the fourth is too cute miners, while the only thing they were subtracting 3.40. Finally, we&#x27;re asked to scale this polynomial so that p three bar is minus 1202 and one. And of course, we see that in our particular case, be three bodies given my minus 1.220 months 2.4 and 1.2. So all we need to do is to scale this vector by one over 1.2, which is exactly 5/6. So if we divide by 1.2, we get the scale. Vector is carefully. Normal is 56 off a cube minus 17 6th off T. And in fact, this bully normal where we take the vector, Sit through it with the bar. He&#x27;s exactly minus 1 to 0 minus two and one has wanted.

Runpeng Li · Answer

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&#x27;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&#x27;s t l Q. So that&#x27;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&#x27;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&#x27;re done for the first part because I our this relation and second relation. Then we can conclude that he&#x27;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&#x27;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&#x27;re plugging t which gives Dr Inactive three negative one, one and three. Then we&#x27;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&#x27;s it.

Arsalan Afkhami · Answer

they flew me out. We have is a zero plus a one X plus a two x. We need to find the victor. That&#x27;s to be even, or the basis, which has three elements one plus x x x minus one Andi one plus two Expert, too Victor Miller before has the From I c one. C two c three to obtain it. We quite the front side. Just you want for it First element setem by second C three by body said so here we re righted a C one plus c one X plus C two, uh, x two. I&#x27;m just multiplying everything minus C two x plus C three bus to he three x or two and the finest Soon our powers off X here it&#x27;s too. And you have, uh, ex then so many Carver. So if we write it again, you will have a system of equations. That is, this has three equations. One first one. These C one plus C three is a close to, uh hey zero as against you here and C three plus C one physicals to his ear. I don&#x27;t seem like for the rest. We&#x27;ll have C one when you see to this close to a gun on C two plus two c three, close to a to then we can write the augmented month metrics cursed by me to these sent if equations the cheese one year show you here is 10 Foresee 21 for C three and a close to a zero. So 10 one most a zero one. Anything you want, Geo. You one true. A one. I hate you. So you reduce it to Rochelle on for by using ghosts and elimination. So we have Ah zero Here. Here. Then we get the reduce for as one geo one 01 200 one You can see here. Um they said zero NGO. So one side off metrics zero. And then here you have a zero a to negative aids you plus a one plus a to. And then you have another system equations, in fact. Ah, this against sold Bite back substitution. So if I write it here Newsome precious is have see one plus C three post a one a a zero and see two plus two c three Those two a to then see three easy cause too Ezio bus a one plus a two. But you can do here is to replace C three to get C two then and also replace C three to get C one. When you&#x27;re done replacing gets the director we&#x27;re looking for, which will be P of X relative to be because to to a zero minus a one, I say to to a zero nice to a bon our state to zero plus a one plus a two. And this is the picture for this problem.

Problem

Let $V$ be a vector space that contains a linearl…

Problem 34 Medium Difficulty

Answer

$\left\{p_{1}, p_{2}\right\}$

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

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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$\left\{p_{1}, p_{2}\right\}$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Kayleah T.

Joseph L.

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

Linear Algebra and Its Applications