in-exercises-11-and-12-find-the-dimension-of-the-subspace-spanned-by-the-given-vectors-leftbeginar-2

Question

In Exercises 11 and $12,$ find the dimension of the subspace spanned by the given vectors.
$$
\left[\begin{array}{r}{1} \ {-2} \ {0}\end{array}ight],\left[\begin{array}{r}{-3} \ {4} \ {1}\end{array}ight],\left[\begin{array}{r}{-8} \ {6} \ {5}\end{array}ight],\left[\begin{array}{r}{-3} \ {0} \ {7}\end{array}ight]
$$

Runpeng Li · Accepted 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.

James Chok · Answer

Okay. In this question, we will find the basis of the subspace span by these vectors. So this is a quiver tow asking what is a column space off this matrix when you just squish all of these colored pictures together? So this column back the girls yet some guys here goes here, goes here, goes here. Does it really matter what water it is? But but I&#x27;ll just take a funeral. So then the first thing that we have to do is to find to convert this into right echelon for so the first couple off steps that will do is take a road to you, add one that she wrote to will take Row three and add two tons of road one. In that interview, I three three and then we&#x27;re full. You was tracked. Why? It has wrote one. Bring that into road for so performing the steps 120 minus 131 plus one gives you zero to my street. Get it minus 10 clothes to give to four months. One gives you three, three months. Eight gives you negative five now very three plus words. You so negative Two plus two gives you zero four minus one gives you 30 My sixties Negative six negative seven miles to give me make you mine and nine plus six kids. A kitty. Okay, now rough Full minus 51 So five minus 50 six minus 10. Giving negative pole eight minus zero gives you eight. Seven plus five. Getting too. No. Sorry. Well, and by minus 15 gives you maybe 10. So now, right oppressed That perform is row three and add three. Title wrote to going down into row three and wrote full will subtract four tunnel road to connect to work for So we get 1 to 0 minus 130 Advice 123 Negative five. And you&#x27;re so right. Three plus road to sew. Three plus three miles st zero to negative six plus six is your negative mind plus nine zero 15 minus 15. Now I wrote for my forward to Is there a negative? Four plus four Q zero eight minus eight is you 0 12 miles. Talk gives you zero, but this is a regular for my sake. Zero on dhe negative. 10 plus 20 gives you 10. So you see, we only have three columns, right? Yeah. So how How homes waste is the space to span by These three letters stood in the basis for this subspace. Ah, one minus one minus 25 23 months. 16 and three. Negative, 89 and five.

Ashley Boni · 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.

Doruk Isik · Answer

and this problem, we&#x27;re given a sending electors knew where to find the basis for this that so let&#x27;s form our system for big from the given lectures. Endless side of this matrix site is our system a CZ you can see hey has four rows and by colors. So it means that a mite hair enough given elements for Pete Rose, since the number of problems is greater than the number of rows. So we know that this system will spend the space. Um, next thing to do is to find the reduced role for or by this system and that Is he going to 1000 0100 000 Sorry. 0010 Negative one negative. 300 Negative to link two by two and zero. Since we know that this settled vectors, I&#x27;ll spend the space Don&#x27;t be basis. Dundee column Give its columns will be the basis for the system. Hey, what are you giving calls? The 1st 1 the 2nd 1 and the 3rd 1 will be our give. It holds so many faces where we set a will be the first told him off a 1001 The second column Obey negative to one negative one. And her column? Um A, which is six negative one, too.

Daniel Pezzi · Answer

hello for this question were given four vectors. W is equal to 312 V one is equal toe 10 negative. One V two is equal to 213 and V three is equal to 4 to 6 and are asked to answer three questions. This question is gonna be testing your rotational skills a little bit. So the first question were simply asked If w is in the set v one v two v three. The same is asking is w the same vector as V one V two or V three and we could immediately see that it is not. The answer to this one is no. And you could just check that the number in the first slot here is three. The number of the first lot of V one V two and V three is not three. So there is no way that this factor is the same as these three. It&#x27;s not asking if this is a scaler multiple of these three or a combination or anything like that. It&#x27;s simply asking if w is one of these three vectors. Now we are asked how many vectors are in the span of V one, V two and V three. And this is again kind of a tricky question because the answer to this, unless all three of these vectors was zero, is always going to be that there are an infinite number. So there&#x27;s an infinite number of vectors without even having to test without even having toe to think about what the dimension of of the set is or anything like that. Because we have non zero vectors in the span and we have an infinite number of Constance to multiply those that factor by, um, in case you&#x27;re wondering what the dimension of this set is, we could just eyeball it and see that V three is two times V to you. Just multiply through and see for yourself on day. These two vectors are linearly independent, so the dimension of the span is, too, and it&#x27;s automatically a vector space because of the definition of span. But the actual, like number of physical vectors in the span is going is going to be infant, and that would be true for one dimensional subspace, two dimensional, subspace, 17 dimensional subspace three only exception is a zero dimensional subspace, which is just this set. And finally, we&#x27;re asked if w is within the span of these three vectors, So can we write W as the linear combination of V one? Um, plus the two I&#x27;m going to ignore the three first. Second, because we&#x27;ve established the V two and V three of his constant multiples of each other. And the answer is yes. Um, the constants, in fact, are, uh, the easiest ones. It&#x27;s one and one, because if you add one and two, you get three. If you add zero and one, you get one. And if you add negative one and three, you get to. So, yes, W is, in fact, in the span of these three factors, and that&#x27;s it.

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Determine the dimensions of Nul $A$ and $\mathrm{…

Problem 12 Medium Difficulty

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Linear Algebra and Its Applications

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

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

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