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3. (10 points Find the order of the given element in the direct product . (u(: ; ; eZw ((; 31) s) eV xSxZu (Hint: V is the Klein 4-group discussed in Section of the...

Question

3. (10 points Find the order of the given element in the direct product . (u(: ; ; eZw ((; 31) s) eV xSxZu (Hint: V is the Klein 4-group discussed in Section of the text.)

3. (10 points Find the order of the given element in the direct product . (u(: ; ; eZw ((; 31) s) eV xSxZu (Hint: V is the Klein 4-group discussed in Section of the text.)



Answers

Determine the component vector of the given vector in the vector space $V$ relative to the given ordered basis $B$. $$\begin{aligned} &V=\mathbb{R}^{3} ; B=\{(-1,0,0),(0,0,-3),(0,-2,0)\}\\ &\mathbf{v}=(5,5,5) \end{aligned}$$

We need to determine the component Victor me, the cheese by by by really really related to the given or the basis which I write A three a shepherd points negative. 10 You hero. You negative three and then the negative too. Here on the picture, we're looking for who we like. C one c 233 to get it right given Rector, because to see one by by the first point plus see to multiply by the second point on C three supplied by 1/3 punch. So here we have X. Why is it on X? Is a call to see one fly by the except the first prime seating life by the tits off this second point and see three multiplied by the X off the third point on this is very easy and to solve so system for equations would be negative. See, Ron C 20630 close to five and then for y Singh. Likely we will have, um see, one must supplied by why and so on. So here, zero foresee to also zero for the and for C three. He's negative too. So I think it is too much by C three is a call to five. And then for the set component, you'll have zero for See Juan. Negative. 35 I see two because to five I've been easily recanted their money. She won C two c three on that will give us the director lifted too. The order basis. Be course to needed. Five in five, do I? Three five, huh? So this is the find an answer.

Get that. Ah, misspoke solution at the end of Ah, my security and get so in this question, they want us, Teoh, take the specter. It lives in our 37 only three dimensional vector and right in terms of the ordered basis on the right, and we're gonna assume their basis. Check that. But this question writing this in terms of this basis boils down to solving a system of the nuns. Okay, You want to take all their ex components, multiply it by some unknown, and it should equal negative nine. Okay, well, the same thing should be true for the white coordinates and the Z coordinates. So we can actually write that in terms of a matrix vector problem, and then use Gaussian elimination. So we're gonna do a augmented matrix. Ah, In order to do that, we're going to eliminate some coefficients in this matrix with our row operations. We're going Teoh, subtract row one from row three again. So this zeros out Very nice. But now let's make use of our road to, and we'll subtract that from row one can't simplifies even nicer, but will also multiply it by two and add it so the bottom wrote zero this out and that becomes a zero here. We still have a negative one, but this becomes a three That doesn't look is nice. So let's multiply everything here by a negative one. So this becomes one. This becomes a negative three, and I think unjust. Easily Reinterpret this. This is just the third coefficient is negative. Three is just the second coefficient as one. And while we interpret this is an equation will have a negative 10 plus two times are there done known, which was gamma, uh, negative time plus six That's actually negative for Can't forget that negative there. So in terms of our I get the area C new basis, we have negative for who is the racing comma? One comma Negative three. You can do some quick arithmetic in your heads and note That's true. I'm going. So we need to get negative night Were negative for plus one negative three. Okay, track six. They really negative, man. So that's 70 working out. You can verify

We need to detain Mind the company and Victor off me. Be cheese. Negative. Nine one minus eight. Let it to be given ordered basis. Here you have X. Why Zet and then you, Victor, will be like I see one. See to C three. So I write the equation, which is C one here. The excellent, uh, no reply by the first point off the basis point, which is one geo on one plus c two don't supply by one one on negative blanc plus C three by two geo one. So what we need to do is to write a system of equations. So here X is a call to see one. Bye bye. The first points X c to live by. Wait a second points, ext on C three months by by day for third point's X. So something like this negative nine. Because to see one plus see to plus to see to And then we'll have similarly for see to which will be close to one. And for C three you, for the Senate question will have C one minus c two plus C three close to negative eight. So to solve this, you have see to. We can replace seats here. And then he writes another simplified equation, which is C one. It goes to minus seven, minus C three and then this. Here, replace, See one. And we also have C two front here. So we will have negative Ni Nichols toe minus seven, minus 33 plus one plus two c two on. Then if you simplify this bill writes C three course to minus three and then by replacing C three here you have see one Costa minus four and we already had C two. So the victor me related to the order basis speed will be minus four one minus three. So this will be the answer to this question.

Hello there. For the following exercise we need to consider three factors. The factory. You the vector V. And the vector W. And taking these vectors. We need to calculate the following cross price this you cross product of the yeah, bacterial team of the cross product of be with W. So let's let's first calculate the cross product between V. With all of you. You know that this can be calculated using the determinant. So he would put the V vector the components of the vector zero to minus three. And here we put the components of the vector W. So too six and seven. Then this can be easily calculated. So in the ice component we're going to take the the terminal of the minor. Well basically it's a minor of this uh this soup matrix And this is a 14 last 18 in the components. So I'm going to put in the different station as a group. Yeah. Then for the next terms of the factory we need to consider now this part, the J. Part. So we eliminate this this column and this row and we have the minor. The minor is going to be just ah 6 -6 -6. And finally for K we need to take these components, eliminate this road and this column and this row and we have the minor here. So that Minor is equals 2 -4. And that's it. So the cross product between the And of you. Is it close to 32 -6 and -4. Right now we need to take this cross product and then take another cross product. But with you, So that's the second part. So now we are going to calculate what we need v cross Dav you and this result into the cross product between so I Jk here we put 3-1 and here below we put the result of the previous part, So 30 to -6 and -4. The determinant, there's sort of, this determinant Is equal to 14 -20 and -82.


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