in-exercises-7-10-let-w-be-the-subspace-spanned-by-the-mathbfuprime-s-and-write-mathbfy-as-the-sum-2

Question

In Exercises $7-10,$ let $W$ be the subspace spanned by the $\mathbf{u}^{\prime}$ 's, and write $\mathbf{y}$ as the sum of a vector in $W$ and a vector orthogonal to $W$
$$
\mathbf{y}=\left[\begin{array}{r}{-1} \ {4} \ {3}\end{array}ight], \mathbf{u}_{1}=\left[\begin{array}{l}{1} \ {1} \ {1}\end{array}ight], \mathbf{u}_{2}=\left[\begin{array}{r}{-1} \ {3} \ {-2}\end{array}ight]
$$

Ashley Boni · Accepted Answer

So you want to write the vector? Why? As a sum, um, a factor And w and a vector orthogonal to w or W is a span off the to use. I was So to do this when you look at, uh, the earth orginal projection of why onto this band of you wanted you two to get the vectors in W s o. The formula for that is a vector. No more protection. Why hat? We project onto one. It&#x27;s a top product and we add a projection onto you too. So first, let&#x27;s find this vector. Eso Let&#x27;s find over dot products you have. Why dot you won Does he call to negative one plus four plus three. So we get six. Uh, you want a product with itself is just gonna be one plus one plus one, 43 So those first coefficient six over three, which gives us two, uh, for the next one. Why dot product with you too, is one plus 12 minus six, which gives us 30 minus six or seven. So you two dot product with itself one plus nine plus four. That gives us 14. It&#x27;s our second coefficient is seven over 14 or one. So let&#x27;s go ahead and multiplies coefficients in. So to find you one gives us 2 to 2 and 1/2 multiplied by a factor. You too gives us minus 1/2 three halfs and get up on so adding use together to minus 1/2 iss Very halfs. Four halfs and three half&#x27;s gives us seven halfs and two minus one is So this is director. Why hat and so to find her other vector, we just need us attract. So we&#x27;re going to take, uh, why Ann&#x27;s a track. Why hat? So we get what we need to add. So why I had to get back to the vector y. So why minus why hat? So we have negative one minus the halfs. Four minus and a halfs and three minus one, which gives us negative five house more minus 7/2 sis 1/2 and three minus Want us to so we can write Why, as the son of this vector and why had

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.

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.

Linh Vu · Answer

in this question. Even if you Miko Ju minus 123 and with the vehicle did you June 11 here. We want to decomposed you judo to be president. Er and whenever the is better, naturally better v here, we gonna be you equal to the projection off the weather year on V and therefore we should get my final computer. You don&#x27;t fee? I&#x27;m a baby. Don&#x27;t be times better, V and you don&#x27;t be. We get me going to, uh three The riding by the gnome under vetiver Vico Judah far plus far less wondrous ones will be six and now will be to 11 And there was a technical Juna 1 1/2 1 half on here. When we&#x27;re gonna be, we can find a renter. And here we a coach in a humanist be Do we get a coach in Murmansk? 123 minus. Don&#x27;t be Now, do we get Nico two? Yeah. Begin minus two to minus 1/2. So we go Teoh tree, Enter 23 minutes 1/2. So a cogent of 500 to therefore again the better be equal to 1 1/2 1/2 and about the on an echo to want us to treat that you find out that you system demented you it could you be plus and

Foster Wisusik · Answer

Okay. This question wants us to split up our vector You into components parallel and perpendicular to V. So for the parallel component we know that&#x27;s just given by the projection of you onto the which is just the dot product of U and V, divided by the magnitude of the squared times V. So now let&#x27;s just plug in for all of these. So for our doubt product, that&#x27;s negative. Two plus two plus three. So three and then our magnitude of these squared is four plus one plus one or six. And then our vector V is 211 So then distributing that 1/2 we get are parallel component is one 1/2 1 half. And now that we have this, we confined our perpendicular component just by subtracting the parallel component off of our original factor. So you is negative. 123 and you parallel is 1 1/2 1 half. And then doing this component wise we get you perpendicular is equal to negative too. Three halves, five halves

Linh Vu · Answer

because even if you go junior for three and the letter V go to 11 they want you decomposed. A better new into the B plus am Where be here? We burn out you the big so honestly done that here will be the projection off the with a new on the Veta V and we can get this one will be the new doctor we offered of the doc fee times the Veta V Then we should we have heard you don&#x27;t vehicle Judah for plus three number one squared plus one square in +11 Then we get Echo just 700 to 11 We get equal to seven under, do 700 to and then also weaken Ghenda better. And because you echo to the peoples and their fund and we could to the human speed. And then we get echo Judah for three minus the 700 to 700 to don&#x27;t get a go to you Never get, uh, eight within going half and isn&#x27;t again this could you? The man that&#x27;s gone half so therefore again the vet the be equal Do the 700 to 700 Jew and a better and Nico Judah 1/2 and when it&#x27;s won half so stop, uh, you we could, you know, be plus and

Problem

In Exercises $7-10,$ let $W$ be the subspace span…

Problem 8 Medium Difficulty

Answer

$y=\left[\begin{array}{l}{\frac{3}{2}} \\ {\frac{7}{2}} \\ {1}\end{array}\right]+\left[\begin{array}{c}{\frac{-5}{2}} \\ {\frac{1}{2}} \\ {2}\end{array}\right]$

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

Linear Algebra and Its Applications

Lily A.

Catherine R.

Kristen K.

Joseph L.

Vectors Intro

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$y=\left[\begin{array}{l}{\frac{3}{2}} \\ {\frac{7}{2}} \\ {1}\end{array}\right]+\left[\begin{array}{c}{\frac{-5}{2}} \\ {\frac{1}{2}} \\ {2}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Kristen K.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Catherine R.

Kristen K.

Joseph L.

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

Linear Algebra and Its Applications