in-exercises-1-4-write-mathbfy-as-an-affine-combination-of-the-other-points-listed-if-possible-mat-4

Question

In Exercises $1-4,$ write $\mathbf{y}$ as an affine combination of the other points listed, if possible.
$$
\mathbf{v}_{1}=\left[\begin{array}{l}{1} \ {2} \ {0}\end{array}ight], \mathbf{v}_{2}=\left[\begin{array}{r}{2} \ {-6} \ {7}\end{array}ight], \mathbf{v}_{3}=\left[\begin{array}{l}{4} \ {3} \ {1}\end{array}ight], \mathbf{y}=\left[\begin{array}{r}{-3} \ {4} \ {-4}\end{array}ight]
$$

Gideon Idumah · Accepted Answer

for this question were given the wan to be deep also Won soo zero vite suit to be recalled to true minus 67 under Envied three to be equal to 43 warn why equaled so minus 34 minus four. I want to write Why, as a fine combination of the other points. So to start waits we computes the translated points The transit of points v two minors v Want, which is won&#x27;t minus feeds on seven. You do a victory minus v Want, which is 31 Juan. And they do why miners ve worn, which is minus for LSU minus four. Next we find skill Oz seats Sue century Such nuts. Seats to veto Miners v. Juan plus seat Ori Victory Miners v. Juan. You want to know why miners v one We do this by we reduce This implies are reduced The augmented metrics with this points as the cool. Um so the matrix will be one minus eads seven three won&#x27;t warn minors for sue minus full. You want to put this street land here? Just tow, give the points but it doesn&#x27;t matter when you roar would use these. So when you reduce these, you&#x27;re gonna have 10 minus two over five zero Juan minus six over 5000 So we see that difference. Consistence under this is like seats sue on dhe seat during Sami&#x27;s. See two equals toe minus two over five on. Then sit. Three quarto Minor. Six over five. So this implies that seed so equal to minus two over five on. Then see the records of minor six over five when you plug us into the question of why miners view want equal toe citrus minus two over five off Vizio Miners v Juan minus that city theory victory Miners we want when you solved a CZ you off white quartz fits in over five. We want minus tsu over five V two minor, six over five. The three.

Michael Jacobsen · Answer

for this example were provided with the vector y, which is 53 Merkel here is to write it as 1/2 in combination of the one through before. To start off this process, we&#x27;re going to calculate some vectors of the form be to my SV one which in this case is going to be equal to negative 30 We&#x27;re going to next calculate V three, my sp one, and in this case, we obtained the vector negative 12 Next, we have to calculate V four minus V one and we find that before my eyes V one is equal to 25 And lastly, we consider the vector why we need to calculate why my SB one next and this is now equal to 41 So now we have four vectors to consider. And what we&#x27;re going to be checking is if there exist waits C one, c two, c three and C four that toe one such that C one times V to my as V one plus C two times V three mice V one plus C three times before my SV one equals Why minus the one If we can solve for C one, C two and C three in this linear combination for white mice, V one, then by the theorems in this section we&#x27;ll have that. Why is an effin combination of V one through before and then we confined the specific waits for that combination. So let&#x27;s start by solving this system by use of an augmented matrix from the vector equation. We have the two mice. V. One is our first column, but that&#x27;s negative. 30 The next column is negative. 12 then 25 and we&#x27;re augmenting with the right hand side. Why, minus few one. Which we have as for one. So we have to roll. Reduce this matrix into its role reduced echelon form. And once we do so, the form is 10 negative. 1.5 negative, 1.5 for the first row, and in the second row we have 12.5 and 0.5. So these columns correspond to C one, C two and C three. And don&#x27;t forget that we&#x27;ve augmented. So now we can solve for C one. C one is going to be equal to positive 1.5 times C three minus 1.5. Next see, too, is equal to negative 2.5 C three plus 0.5, and our free variable is C three soldiers, right? Si three equals C three. To represent that, there&#x27;s no restrictions, so we can choose whatever value we have for C three. Let&#x27;s set C three equal to zero. In that case, if C three is equal to zero, then we find that C one is equal to negative 1.5 and see to as soon as we put zero. In place of C three gives us 0.5 altogether. So now we&#x27;re ready to write the linear combination. We have first negative 1.5 times V tu minus V one plus C two, which is 20.5 times V three mice V one plus C three. But we set that to zero times before minus B one, and this is equal to white minus V one. Now if we distribute into each of these groups, we have negative 1.5 v tu minus 1.5 V one plus 0.5 V three mice 30.5. The one equals Y minus V one. Next, add be want to both sides and combine like terms. And we find that why is equal to two V one minus 1.5 v two plus points five v three plus zero times before and to determine if this isn&#x27;t a fine combination. Considered these weights listed here we have two minus 1.5 plus 0.5 is equal to exactly one. And so why is now an end fighting combination of these listed vectors, with the given waits highlighted in purple?

Michael Jacobsen · Answer

in this example were provided with the vector y from our three as well as three other vectors. Be 13 V three a nickel Here is to determine whether or not why is an F in combination of the vectors V one, V two and v three. So our standard method in making that determination is to first folk focus on the vector V one and what we&#x27;re going to be calculating are V to Biosphere one to start out, and for this particular difference, we get altogether 33 and negative three. Next, go to the vector V three and calculate the three by minus V one as well. In this case, be three mice. V one becomes seven, a negative three and a five. Lastly, we&#x27;re going to calculate as why minus V one as well. And this results in the Vector 20 zero and four. Now we need to explain why we&#x27;ve considered these particular vector differences. If we form the make or the vector equation C two times the first difference v tu minus v one then plus C three times the second difference V three mice v one set the result equal to why my as V one. Then we can determine that the vector Y is an and fight combination of the 13 v three if and only if this victory equation is consistent. So to determine consistency, let&#x27;s start by going to an augmented matrix. So the unknowns air sea to sea. Three. So the first column is V two Lions V. One, which we found to be 33 negative three. Then take the next column. Seven. Negative three and five and the right hand side is going to be white mice V. One So augment to this matrix with 20 zero and four. Next, we need to perform Roar operations until we&#x27;re in roll reduced echelon form. And the reduced echelon formed for this matrix is 102 012 and the last row contains nothing but zeros. Now for this matrix, let&#x27;s recall that we&#x27;ve augmented, so the right hand side are the constants, and the first column corresponds to see two. And the second column corresponds to see three. So this implies that C two is going to be equal to two. In the second row. We see that si three equals two and the third row gives us no contribution. So now we have values for scale. Er&#x27;s C two c three. Let&#x27;s re express. This equation will have to times V to my SV one plus two in place of C three times V three minus V. One is now seemed to be equal to why minus v one, and this implies already that why is an F fine combination for the one V two and V three? Since the equation that we considered here is consistent, it doesn&#x27;t tell us what the actual scaler czar for their fine combination until we saw for why. Let&#x27;s take a initial step by distributing. So we have two V two mice to V one plus two V three mice to V one equals Y minus B one. Then add the wonderful sides collect like terms and we see that why is equal to negative three V one plus two v two plus two v three. So we claim this is the specific at fine combination bliss. Check here. When we look at all of these coefficients, they must sum to exactly one, and that is the case. So we know how to write why, as an F in combination of these factors listed here

Gideon Idumah · Answer

two rights. Why, As on a fine combination Off the points v Juan, equal to 11 v to equality So minus one suit Victory records of 32 we&#x27;re why equals five and seven. So the first thing we do is we computes the translator points computer translated points you r V two Minors v wall, which would be minus two comma Juan and then v three miners v wall, which is two on one on the white Miners v Juan, which is four on six. The next we find skill. US seeds to commerce secretaries Such nuts. Seats to multiply by V two Minors v Juan Lost City three v Theory Miners view Want equals white miners V want So we do this by we&#x27;re all reduce The augmented metrics have in this point of school. Um, so you have minus 21 soon wrong for six. When you reduce this matri&#x27;s you should get so ro reduce. You should get 10 sue 014 So this implies that my sit suit with Sue on my seat table before, So this implies that&#x27;s sits who quartered soon London City. Three quarts of four. So this implies That&#x27;s why miners we want records. My city is not Sue Veed Sue Miners view on my sitter is four victory Miners v Juan. When he saw for this you have the white equal to minus five of you won lost to V two plus for the 30.

Foster Wisusik · Answer

Okay, This question wants us to test if the&#x27;s points are Colin here. So to do that, we have to check if the area determinant is zero. I call this the area determinant because it tells us about the area of a triangle. So if the determined zero, it means our triangle has no area, so the points must be on a line together. So that just means we gotta evaluate a determinant. So we have to negative for six because the axe is going column one and then their corresponding wise going column, too. And then one&#x27;s going. Com three. So let&#x27;s pick column three to expand along. So our determinant is just deleting their own column. Negative for for six negative three minus to negative 1/2. Six. Negative three. Plus it&#x27;s late the road to lead the column to negative 1/2 negative for four. So now we just got to expand these two by twos to get well. Our first determinant is 12 minus 24. Then we&#x27;re subtracting negative six plus three. Then our third determinant is eight minus two. So then let&#x27;s do this out and we get negative. 12 plus three plus six and this is equal to negative three, which is not equal to zero. So since the determinant is not equal to zero, the points are not Colin here.

Gideon Idumah · Answer

other points of view. One equal to 131 v tsu equal to zero minus one minus two V three equals Who sue five. Soon Andre V for records. Who? 350 So Chocolates V two minors view on equality so minus one minus four minus three victory Miners view Want equal to 1 to 1. And there were cockle. It&#x27;s V four Minors we want equal to who gets to sue minus one. And I want to check if this is dependent on independent So we&#x27;re we&#x27;re reduced the metrics form when you&#x27;ve turned out since the images to get minus one minus four minus three. I want to do want soup so minus one when you&#x27;re already is these you&#x27;re gonna off 100010001 We can see that this is linearly independence. This implies linearly or linear independence. Why? Because every cool, um, ask the privates. So you have a private. You have a private pilots here, so this implies are are fine independence

Problem

In Exercises 5 and $6,$ let $\mathbf{b}_{1}=\left…

Problem 4 Easy Difficulty

Answer

$y=\frac{13}{5} v_{1}-\frac{2}{5} v_{2}-\frac{6}{5} v_{3}$

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

Linear Algebra and Its Applications

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$y=\frac{13}{5} v_{1}-\frac{2}{5} v_{2}-\frac{6}{5} v_{3}$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Heather Z.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Heather Z.

Joseph L.

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

Linear Algebra and Its Applications