in-exercises-11-and-12-find-the-closest-point-to-y-in-the-subspace-w-spanned-by-mathbfv_1-and-mathbf

Question

In Exercises 11 and $12,$ find the closest point to $y$ in the subspace $W$ spanned by $\mathbf{v}_{1}$ and $\mathbf{v}_{2} .$
$$
\mathbf{y}=\left[\begin{array}{l}{3} \ {1} \ {5} \ {1}\end{array}ight], \mathbf{v}_{1}=\left[\begin{array}{r}{3} \ {1} \ {-1} \ {1}\end{array}ight], \mathbf{v}_{2}=\left[\begin{array}{r}{1} \ {-1} \ {1} \ {-1}\end{array}ight]
$$

Ashley Boni · Accepted Answer

so we want to find because this point toe why and the subspace w which is the span of the vectors of you want to be, too. So to do this, we know we need to find the orthogonal projection of why onto the span of you want to everyone a beat. So we have before meal for that. And that is why hat equals why dot b want over the one dot product with it. So Vanity one and some early with vegetable. So let&#x27;s go ahead and find these dot products off to the side. So why not? Product would be one gives us three plus two minus one most 26. So we have 26 plus five is 31 minus one is 30 be one. That product with itself gives us one post for close one plus four, which gives us 10. So this first coefficient just 30 over 10 or three. So next, Uh, why dot product will be too. So we have minus 12. Find this one plus zero plus 39 so we get 39 minus 13 because it&#x27;s 26 and gratitude out product with itself, you get 16 plus one plus zero, which gives us 26 as well. So second coefficient is just one. So now if we multiply these coefficients through three times 31 get three minus six minus 36 plus Ah, one times we too. So we just copy down too. And add these together. So three minus four. Negative one. Make it a five negative three and I So that&#x27;s the vector that we won&#x27;t.

Runpeng Li · Answer

here we have the four Rector&#x27;s. It&#x27;s been such a subspace. That is, 12 zero. Think of 34 one and noted eight, six, five and negatives 30 seven. Excuse me. So, in order to check whether this is four vectors, are you nearly end end? One approach I would recommend is to apply the gushing illumination and check the reduce station warm to see Hi. If there is an theories and then travel and trivial solution. If there is, then that means the&#x27;s four by four vectors are leaner depended. So we&#x27;re done. So Okay, so to do the cash indignation. Oh, just right down First on the first page here and now we&#x27;ll we&#x27;ll keep going to the second page. Okay, so we first views. Second row, plus two, two times. The first road myself. The first rule. So that is fun. Next three, activate connective three, 23 And so the second rule will be zero. But first century and two Sorry. Connected to the second entry. And negative 10 with 1/3 entry, connective six. Or, uh, third entry for the fourth entry. And we&#x27;ll keep the the last roll. Okay, Next, we will try to reduce the the last roll here. We usedto third row plus twice out the third row. Plus, um, that&#x27;s the second rule. Okay, so first row were not changed. So that is one negative three reactivate a negative three. And so second rule was still being active to connective 10 and neck of six. Well, that&#x27;s certainly the third row. We have zero first and cereal with 1/3 entry and two times 7 14 minus six iss eight. All right, so we don&#x27;t need we don&#x27;t even we don&#x27;t even need to redo. Keep reducing, because right now we can We can know whether there are where there is a nontrivial solution. So we first have um x one minus three, x two plus eight Sorry, minus eight. Minus eight x three finest three x four minus three x for is zero. And we have negative too. Thanks to minus 10. Next three finest six x four. It&#x27;s zero and eight times. Explore is always zero. So export zero. But we still have X y extracts three. Right. So first thing we have next one minus three x d&#x27;oh. Minus aid x three. It&#x27;s zero and connected to x two minus 10 X tree. It&#x27;s Darryl well, ex for zero so we can find from thes you questions that x two x three and, um, yaks. Three is a free, bearable, and X two can be determined by X three and X one can be determined by X two and x three. So that means there&#x27;s there exists athletes and nontrivial solution. So that means these four vectors Do you want to three Before are we nearly dependent? Independent dependent. So that implies dimension off the subspace it&#x27;s for.

Ashley Boni · Answer

so from question 12 we already know the orthogonal projection of wise onto the span of the wanted me to was equal to the vector minus one. Finest five Negative. 39 So now we want to find the distance from why? To the given suspect the given subsidies. So we need to first take why? Minus the projection. So we have three minus one 1 13 minus minus one minus five minus 39. So yet four, uh, negative. One plus 54 one minus Nick Siri&#x27;s for 13. Last night is for so we get the vector 4444 So now we define the distance from why to the plains banned by if you want to be too. So we just take the norm of this difference We just found and square it because that&#x27;s a formula, for instance, So the norm is equal to square root of each component squared. So for squared, close for squared plus four square puts four squared and then we square this so square root goes away and we just get four times 16 which is 64. Um, so we end up with the square root of 64 which is eight

Gideon Idumah · Answer

No, let&#x27;s recall That&#x27;s recall. That&#x27;s X is in no all of a if a times eggs he called zero. Now, in our example, with all that it is a four by three ministries. So what would be the size of X? You can see that ex us to be a three by one matrix because this true must align. So this tells us that&#x27;s for no off, eh? So being is subspecies off our P p us to be a tick or 23 because the vets are in null off aid isn&#x27;t too good by one vector. Saddam is none off a for none of it is suspicious of our p p has to be recalled to three. Now that&#x27;s off the other part of the part of the question. Now recall that&#x27;s B is in cologne. Space off, eh? If there&#x27;s an eggs, days, eggs, such dots a times X records to be right. No, this is a four by three matrix. This has to be a three by one matrix. Vento. Then what would this be? So this is for by through Constant Terry. So this has to be a four by one. All right, so that implies that full Coolum space off, eh? To be his subspace off our cue que us to be because B is in column space on B is for buy one mentions some us be us to be cute. Bastard cost of four

Ashley Boni · Answer

So we want to write. Uh, why as a sum of vectors one a w and one orthogonal of W where w is just, uh this pan of all these use. So first, to find the one N w we need to project why they were on Waned of project. Why North, Ogden Lee, onto the span of you want you to endure three. So to do that, we&#x27;ll take our projection formula and project. Why? Onto you won. You too on you three. So let&#x27;s find all these DA products off to the side. Why? I thought you want iss three plus four. Plus they&#x27;re a minus six, which gives us one. You want dot product with itself. Make it one plus one. Hello, zero plus one. So I get three. So those first coefficient is 1/3 Next. Uh, why that product with you too gives us hurry. Close zero most five plus six. So eight and six gives us 14 and you, too dot product with itself. Look it 1.0 close one plus one. So three again. So our second coefficient iss 14 over three. Finally, uh, why dot product with you three we get zero minus four. Close five minus six. Negative. 10 plus five gives us minus five and you three dot product with itself again, we get zero plus one plus one plus one. So three. So our last coefficient is negative. 5/3. So now we&#x27;re going Thio multiplies coefficients into all the inspectors. So 1/3 time&#x27;s you won gives us 1/3. 1/3 0 You could have 1/3. Ah, second victor, 14 3rd multiplied to you too. Gives us 14 3rd everywhere. Um, except a zero in the second place. And lastly, negative 5/3 multiplied into you three. So we have a zero 5/3 minus 5/3 and 5/3. So we&#x27;re gonna add all these up. 1/3 plus 14 3rd ISS 15 over three, which is five 1/3 and 5/3. Gives us 6/3 or two 14 3rd Minus 5/3 is 9/3 or three. And lastly, 14 3rd plus 5/3 and 19 over three minus one over three. Gives us 18 over three or six. So our production vector that&#x27;s in W is 5236 So we need to find the one that&#x27;s 4000. All the w. So to do that, we&#x27;re just going to take the vector we want to get, which is why subtract. Why? Hat? So we have 3456 minus that vector we just found, which is 5 to 36 So this gives us negative too. Two, 20 So therefore, um, we can rate why? As a song, Uh 5236 plus minus 22 to go.

Problem

In Exercises 11 and $12,$ find the closest point …

Problem 11 Easy Difficulty

Answer

$\hat{y}=\left[\begin{array}{c}{3} \\ {-1} \\ {1} \\ {-1}\end{array}\right]$

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$\hat{y}=\left[\begin{array}{c}{3} \\ {-1} \\ {1} \\ {-1}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Anna Marie V.

Heather Z.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Catherine R.

Anna Marie V.

Heather Z.

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

Linear Algebra and Its Applications