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HH: 0 pueHzu 1 1 U : 1 I L U E 8ed 1 E 1 81 IH V 5 L 1 L1 1 Il L F L ; 1 L...

Question

HH: 0 pueHzu 1 1 U : 1 I L U E 8ed 1 E 1 81 IH V 5 L 1 L1 1 Il L F L ; 1 L

HH: 0 pueHzu 1 1 U : 1 I L U E 8ed 1 E 1 81 IH V 5 L 1 L 1 1 Il L F L ; 1 L



Answers

(a) $\mathrm{F}_{1}(\mathrm{a}) \mathrm{F}_{1}^{\prime}(\mathrm{a})+\mathrm{G}_{1}(\mathrm{a}) \mathrm{G}_{1}^{\prime}(\mathrm{a})$ (b) $\mathrm{H}_{1}(\mathrm{a}) \mathrm{H}_{2}(\mathrm{a}) \mathrm{H}_{3}(\mathrm{a})$ (c) $F_{r}(a) G_{r}(a) H_{r}(a)$ (d) 0

In this video, we're gonna go through the answer to question number 17 from chapter 9.4. So we asked to find where these vectors x one x two x three Ah, linearly dependent on where they are linearly independence, which FYI is teeth independent much varsity that linearly dependent. So okay, so for them to be linearly dependent would need values off. See one c two c three it such that c one times x one c two times x two plus C three times Next three equal to zero association into values for these x one x two x three then we write This is a system of three equations. So the 1st 1 is just gonna be Ah, well, it's the the first element of each of the equations Time each of the vectors X one experience to text three times by the corresponding um constant C one C to C three. So we're gonna have Well, there's no, uh, ex threes are no component, Maxime, because the top of the next three is zero. So we're going off, uh, ex one next Tuesday, at both of which contain eats the to tease it. So take a common factor out. Get either to t C one plus C two. Ah equals there. Next up, we're gonna have eats the two tea. Close. It was C one that's actually to eat the TT plus C three he to the three tea equal Syria. And finally five. Easter to tee times five C one minus C two equals Sarah. Okay, so comparing the first of these equations on the last of these equations we find that C one don't see too must be equal to zero as you see that because, well, first look in the first equation e to the two tea for any tea can never be zero. So we basically just cancel by its beauty and saying with the bottom equation s So therefore, we have this bit is equal to zero this busy zero for those two to both equal to zero. Then C one and C t o. You can show that by rearranging one of those sub student in the other. Um and you're find this. You wanna see t birthday frequent zero. So therefore I see two is the rocks that this is zero. So then we have to see three times Three tier sequence There we can we can divide by eats and three team because that's never zero for any tea on DDE. All we're left with is C three sequences So far, all t ah the C one c to the T three. Must could only be for this for this equation. Thio be satisfied. Uh, this equation to be satisfied then Theo Dissolution of the Trivial Solution. Therefore the vectors x one x two x three are linearly independent for that tea in any value between minus infinity to infinity.

In this problem we are going to use the properties of mattresses and they're in verses in order to simplify a given mattress expression. Now the given matrix expression is E C inverse whole inverse times E C inverse times E C inverse whole inverse times E. The envious Now in the question it is said that a B C and D. Are all in vertebral mattresses. Now let us begin to simplify this expression. Now, first of all, what we have is easy inverse inverse times, easy inverse. That means that the matrix easy inverse has its inverse multiplied with this original matrix. Now, since the product of any metrics with its inverse is equals identity matrix. Hence a C inverse inverse times a C. In verse will be the identity matrix. I. With this we multiply the remaining two terms A C inverse inverse times a. D. In verse. Since any matrix multiplied with the identity matrix, is that mattress itself? This will be equals to A C. Involves inverse times A. D. In Vegas. Now using the property of matrix inverse, which says that a B hole in verses equals to be inverse. The inverse E C inverse hole in verse will be equals to see inverse inverse times a inverse. And with this we multiply a. D. Universe which are the last two terms of that expression. Now see inverse inverse will be equals to see because the inverse of any inverse is the original matrix itself. Next we have inverse. E be endless. Now, since matrix multiplication is associating, we can group together the terms however we want without changing the order. So let us group together to terms a university. And then we use the property that the product of any matrix inverse is equal to the identity matrix. So inverse A. Will be equal to I. And we are left with C I. D. In most now since the product of any matrix of the identity matrix is the matrix itself and ci is equals to see. And with that we have at the end the in laws, so that means that our metrics expression is simplified to CB in Vals.

They're. So for this exercise we have this vector B. And the subspace dovey generated by the one, V two and V three that are these vectors that are defined here. So basically we need to calculate the Earth a little projection of you on this space to view. And just remember remember this projection is calculated as the inner proud of the vector V. Each of the generators of this subspace dog. In this case the generators RV one, The two and 3. So we need to calculate the we need to calculate the inner part of me with each of the generator divided the score of the norm of the generators times degenerates. So these for the three vectors B two square plus the interpreter of B would be three. B three. Did the square of the norm of B. Three. Okay, so just to remind you a little bit of the geometric intuition of this, is that the view is generated by these three vectors. So what we're doing is projecting we on each of the generators and then some that together. So we want We t. v. one and V three acts as a basis. Actually in this case they are linearly independent so they form a basis for this. Yeah, subspace of you. So we're writing the in terms of this basis. So we're projecting projecting on this sub space. So let's calculate the correspondent values that we need. So in this case we would be one. The product of B would be to dinner product of the would be three. So this is equal two, one half, There is a constitute and this inner product is equal to zero and then the norms. So because this is the cost to zero means that we don't need this term anymore is going to be equal to zero. So we just need to calculate the score of the norms for B. two and B one. So for me, one square of the norm, remember that there is equal to the inner product of the vector with itself. And in this case this result in one and the inner approach of B two square is equal 2, 1 as well. So these are actually military vectors. And then we just need to put all together on the four. So behalf that the projection of the vector B on the subspace, our view, it's equals to 1/4 times 11 one plus the vector V two. That is equal to one, 1 -1 -1. After some. In these two vectors obtain the action solution that is one half times the vector, three, three minus one minus one. That corresponds to their thermal projection of beyond this subspace of you.

Okay. So for this equation, we first wait, defined and operated B as the curative with respect to be so we can rewrite this the stuff of the into something like this. So just rewrite the system for these in this operator notation. Okay, so, um, for now, because I want to use the elimination. Um, so I legal this by question one in the label, The 2nd 1 by question two. Okay, so we're going to pullets and another operator two D pass one two equation one. So we have the first equation will be to t plus one times D plus one, you minus toothy Pass one terms three plus one me because to to the pass one acting on you to the key. So to the process, acting only to be cured with three times eternity. Now, for your question to we use an operator. The pass one acting with you. So we have you pass one times three minus one in the Are you husky past one comes to E pass one v equals to this operator acting on five. So the acting on five p zero because five is the constant. So you know we have five left. Now we can see that to the second term on the sin with different sign in front of it. The muse If I label this by equation three Naval base by question for not three pass for gives us, um, the fooling results The square pastie acting u equals two e treaty plus 5 30 So this is a novel, Virginia. So the, um off you So the correct arrested, um, equation will be ask where Plus, are you close to zero? He has two solutions. So are one in coastal minus one are two recourses Zero. That means the homogeneous general solution for you will be C one e to the minus t plus e to eso here of meat um, e to the zero because it is very close to No, since it's a normal Virginia's are off the we need to consider some particular solution for the right hand side, the right hand side contents and eat with the key. So the corresponding term will be a time CTO key. And who will decide eight after now for the second term is it's ah constant. So we should put a linear function here. Sorry, you should be Ah, eat Hemsky. So he will be decided later. Now I plug in this particular solution used to pee into this of the so is the square plus big acting on your P because to Italy T class 5 30 So live inside since we know your p closed toe eight anti to DT prosperity. So the left inside will be Hey time see to get key plus a time seem to the key for us So let and savvy cost the right hand side Compare the conditions year we have a close to what half and the P equals 2543 So India that we have a general solution for you will be the homogeneous part press the particularly part No. Clear down with the U, But, um, in this system for off all these we have another function we toe figure out. So we have a lot of different choices to figure out the we can just proud you. The results you into either equation wandering question to and the do this Oh, the so far again. But this another solution to this This is a method. So locating the equation one Why I'm going to do is I'm going to rewrite this suit out system for the by. Multiply two for two equation one. So we have twice off E class one acting U plus minus to the minus to me because to thyself be pretty. I just multiply to to both sides and we keep it a second. The equation change, not you fight and the envious to equation up. We can see that since this two operator on a sam with different slide so they can Selves and the way are left with ah minus B so minus V. Because to two attacks to the T plus five, subtract the some off this that will be minus two depressed too acting on you in the minus G minus. So on acting on you so v close to minus two times E to the T minus five plus, um three d acting on you and the class you. Now we are ready to plugging you into this result and that we can directly off 10 V without Soviet me. Oh, these So in the end, the equals two minus two times. See one Italy minus T process See to pass 5/3 key. Just plugging this result into this equation so immediately gets be here


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