find-a-basis-for-the-set-of-vectors-in-mathbbr2-on-the-line-y5-x

Name: find a basis for the set of vectors in mathbbr2 on the line y5 x
Uploaded: 2019-08-11
Duration: 1 min 18 s
Channel: Numerade
Description: Find a basis for the set of vectors in $\mathbb{R}^{2}$ on the line $y=5 x$

Question

Find a basis for the set of vectors in $\mathbb{R}^{2}$ on the line $y=5 x$

Doruk Isik · Accepted Answer

in this problem where has to find the basis or it&#x27;s set up like there&#x27;s all this given line Why is you find so we&#x27;re actually time we sold the system off eggs is zero and we don&#x27;t in This problem we don&#x27;t know are a metrics and it actually doesn&#x27;t matter. Solis This year, ese cooked a be and we have an extractor and that is all thanks. And what? So what we know is this We know that buys you five so we can write. This one is thanks. And then that will be cooked eggs one fire. So this means that we have a times Thanks. 15 is zero. It&#x27;s unique to that should be good to zero. So this system giving here 15 this factor is actually the basis vector that lies on this line wise. You got five minutes

Michael Jacobsen · Answer

In this example, we&#x27;re dealing with the equation of a plane, which is X plus two y plus Z equals zero. The most important thing about this plane is that the right hand side is zero here. That means in particular that this plane is going to go through the origin. And we know now that planes in our three that go through the origin are considered sub spaces in our three or of our three. So in this case, what we might ask is, How could we get a basis for this subspace? We&#x27;ll start off by taking the equation. And solving for the very blacks will have X is equal to negative two y minus Izzy. Then what are the restrictions on the very bulls Y and Z? Well, none of given so we could write. Why equals wife? We like and Z equals Z to convey that there&#x27;s no restrictions. And this also helps remind us that Y and Z are also considered free variables in this situation. Then, with this set up, we can describe a generic vector X that is inside this plane explodes to white plus Z equals zero X is going to be consisting of X, y and Z. So let me put it arrow here to consider this as a vector and to differentiate it from this X. Then the Vector X is going to be described in terms of its free variables, which are Y and Z. So it&#x27;s right first that we have a vector. Why multiplying a constant vector plus a vector Z multiplying a different constant vector and the way we determined that Constance is we take coefficients for or why to place here, which are negative to one and zero since why is missing here? Then take coefficients of Z. They are negative one. There&#x27;s a plus zero z. You can think of this as in either case, there&#x27;s a zero here and a coefficient of one for this. See, and the only stipulation about Y and Z is that why and Z are in the set of real numbers are, as in there&#x27;s no restriction since their free variables. So, looking at this format we see here, this is really just the way we describe a null space, and in particular, this is the same format for a spin of a set of vectors here. Since We&#x27;re dealing with a linear combination where Y and Z can be any real number. So we can now describe this plane as the span of a set of vectors. We say the span of the set Negative 210 We just pulled a constant vector and negative ones. You&#x27;re a one generates this plane. So now we know how to describe this plane as a set of vectors. But what if we didn&#x27;t want all the vectors? We just wanted a basis? Well, we&#x27;re just a step away for a basis we must span. And we already do that with the two provided vectors. And the vectors must be linearly independent, but notice There&#x27;s just two vectors. And from this entry to this injury, they cannot be multiples. And so they are certainly linearly independent. In fact, this process of solving a homogeneous system breaking down in terms of free variables like this and writing love generically in your combination always gives linearly independent vectors. So we can now conclude that the set so don&#x27;t write. Spend in front of it to just take the two vectors. Negative. 210 negative 101 is linearly independent And so since we already know it spans, it forms basis for this plane. So this is an interesting example because this is the type of equation you more likely see in, say, a calculus class were using X, y and Z. We have just one equation for a plane, but linear algebra can still describe this plane.

Ernest Castorena · Answer

get there wants is all the questions similar to before Give in This wrecked it in our two the radio in terms of the basis on the ray that is ordered. So this is the first factor in our basis the second director and the Patriot in The Matrix. This boils down to us. Solving a system of nuns and two variables scale is that I want to buy those rector&#x27;s let&#x27;s see their ex components. Let&#x27;s end up equally fine And there why? Components must end up glowing in negative. 10. We can perform Gaussian elimination on this too easily Read. Our solution was gonna add I&#x27;m just gonna go ahead and add Throw one to road to you So we have to one on five. Here. I&#x27;ll end up with 05 on negative five here. All right, well, I&#x27;m just performing Got the nomination because I can easily read off the solution a bit more. Here, the second row tells us that five beta equals negative. Five. Sophia is equal to negative one, and our second equation tells us to off. Plus, beta equals five. Well, it&#x27;s track that beta, so it&#x27;ll be fire. Subtract negative ones but that&#x27;s plus So it&#x27;s under six and our Alfa is equal to three on. There you have it. Our new vector should be three common negative one. Why is that? Well, five negative 10 is equal to three times negative to common. Negative to you to become a negative, too. Attract one times one come afford. Verify that makes sense like the X compliance we have three times to. So six of direct one. It&#x27;s fire, then the other components. Negative. Six of direct before That&#x27;s negative. Okay, leader that only this gala is necessary to represent our vector in the new basis. Awesome. It&#x27;s always get it.

Arsalan Afkhami · Answer

we need to December in the component. Victor V equals five on Lady Top 10. Listen to you and ordered basis on this new points. Ah, well, the see one on C two to obtain these points your rights The first victor. Because to see one not supplied by the first point off the ordered basis which is to get it to plus see to hear why Inspired by wanna four Here he is our eggs boy. Then we will have a system of equations. Why? Because here. C&#x27;mon, it&#x27;s bye bye to and see tune for by Ron close to five. So to see one plus see to that seem like before did why negative 10 It calls to minus two c one plus for C two and then we can simplify simplify equations. So here it gives us see two which is five minus two c one And then if we replace here these c two here it will give us See Von because to three and then by replacing see Iran here you&#x27;ll get see to he calls too minus one. So director, letting to basis be will be relative to be equals two, three on one. This is the final answer

Jimmy Yao · Answer

we want to find the also the basis for the spender of the sad s. So in science we have the X one and x two. So this is this is X one into this is six to we use Grinch meet person as turkey. It&#x27;s also gonna basis So the one is equal to X one is it is five minus one and two and V two gets X two minus these distracter and we&#x27;re going to compute. Stir in a protective you too extreme A and B one and B one and B one. So for extra in everyone we gets our statistics and they&#x27;re V one and V when we get 30 so over B two will be a similar 11 minors this factor So where you simplify you against this then to the first we did seven money six This one and the second is 1 6/5 will be the level five. The last one is one miners tell the fine it&#x27;s minus seven bullet fired. So according to the branch meat over or hour or so calm paces for their span of us should be one fine minus 1 to 1 that everyone will sly minus several

Jimmy Yao · Answer

here. If one is sequence for one, if two is produced by eggs and stream is because you call Sign X. So I would be one. It&#x27;s the country F one. It&#x27;s because one of the two, it&#x27;s because you have to minors if to be one in a 11 21 so dances the unit Productive of this one will be the NSA. Go down minus one or high. They were to sign ex dear. So this is just er minus. Coastline of eggs evaluates. That&#x27;s the point that this will be zero. So this means you will be too. It&#x27;s ICO to have to speak to sign it. So we think he could too. If the money is see anyone? Anyone being one you One minute. If we were to to to me too. Therefore, greeting batteries. Do you know productive This one B minus pi work through my work too. In there is a co signer. This will be fine. Fly over to in the pyre one to minus pilots. So this will give you two and everyone meet one here. Fly over to my own too. So that is Ah. So the 111 Yeah. We&#x27;ll give you a pie in their three. Me too. The lease is minus. Pi were too high or too That&#x27;s his clothes. Sign. Explain. Assigning there. So this is nothing but Serve one around playing over to imply over to Okay, so you can view it by over two in pirate too. Sign of next in D sighing a big So this is just 1/2 sorry of X. Where your batteries since this to never give you zero. So you equate the street will be coastline ex miners Hi, uh, to India one. Right. So over. Or the also basis there will be one in sighing eggs in the coastline EF minus two.

Problem

In Exercises 13 and 14 , assume that $A$ is row e…

Problem 12 Easy Difficulty

Find a basis for the set of vectors in $\mathbb{R}^{2}$ on the line $y=5 x$

Answer

$\left\{\left[\begin{array}{l}{1} \\ {5}\end{array}\right]\right\}$

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

Linear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Kayleah T.

Vectors Intro

Vector Basics - Example 1

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$\left\{\left[\begin{array}{l}{1} \\ {5}\end{array}\right]\right\}$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Kayleah T.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Catherine R.

Heather Z.

Kayleah T.

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

Linear Algebra and Its Applications