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Question

Let $V$ be the first quadrant in the $x y$ -plane; that is, let
$V=\left\{\left[\begin{array}{l}{x} \ {y}\end{array}ight] : x \geq 0, y \geq 0ight\}$
a. If $\mathbf{u}$ and $\mathbf{v}$ are in $V,$ is $\mathbf{u}+\mathbf{v}$ in $V ?$ Why?
b. Find a specific vector $\mathbf{u}$ in $V$ and a specific scalar $c$ such that $c \mathbf{u}$ is $n o t$ in $V .$ (This is enough to show that $V$ is not a vector space.)

Daniel Pezzi · Accepted Answer

Hello. For this question, we need to do two things with this given subset of our where X and Y are just two constants that are positive. Um, we need to show that given any two arbitrary vectors in this space, uh, the vector edition of those two vectors is also in the space. Then we need to try to give an example of a vector and a constant so that when you multiply the two together, uh, it is not in this space, and this is checking to see if this is a subspace of our two. So for the addition, it&#x27;s typically best just give yourself to arbitrary vectors in here and see what happens. So let&#x27;s say we have one vector with two components, and we&#x27;re gonna add it to another vector with two components where all four of these constants are positive so that these two vectors are in this space. Then when I add them together, even though this seems fairly trivial, you can see that X one plus x two is going to be greater than or equal to zero because x one and X two are greater than zero. And why one and why to are going to be a greater than zero because the individual parts of the addition are greater than zero. And so writing that out formally is the proof that if you take any two arbitrary vectors in the space and add them, you you&#x27;re still end up in this space. Now, for the second part, we can pick any vector in here. I&#x27;m just going to keep it simple and pick the vector one one. And if we multiply, we need toe find a constant C such that this thing is not within that space. And there&#x27;s a new infinite amount of constant you can pick. But if you pick c equals negative one, that implies that these two things multiply together, are negative one negative one just by basic laws of Vector Edition, which is not within the space, because these two things are both negative and that&#x27;s it

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna be selling problem number two of section 4.1, which is, um, on subspace is and vector spaces. So in this problem, we&#x27;re given that W is a set of, um, all order pairs are all vectors that are in the first, first and third quadrants on a graph. So that would just be all over repairs that fit this definition where X times why is greater than equal zero as the first quarter and will only have positive values. And the third quarter will only have negative values on the number lines. So multiplying two negatives would give us a positive and multiplying two positives will give us a positive. So that&#x27;s why this definition fits for this for the set. So part A s us. Whether, um, if a vector us in w then multiplying it by any scaler will also be in W. So, um, to do this, we know that Let&#x27;s say you is x one and X two, and we know the X one times X two is greater than, um, zero. If you multiply the vector by sea like any scaler Ccx um ah si x one time see x two, and we try to see if it&#x27;s greater than equal to zero. We got C square times x one times x two are greater than zero since C squared is a scaler even just divided to the other side. So you know the X one and X two is greater than zero. And think of it logically multiplying, uh, to negative numbers with any scale. And multiplying them together would still give us a value greater than zero and multiplying too. Positive numbers by the same scaler value. See. But, um, any two positive numbers, by any skill of all you see and multiplying them together would also give us a positive values. So if it&#x27;s the definition of X times, why is greater than zero for for this set? And B is asking us to find two specific examples of you and of you that are both in W. But, um, you plus V is not in Dublin, so this is pretty simple. Just pick like the border cases, which are, um, let&#x27;s say you let&#x27;s pick you to be zero comma one, and let&#x27;s pick. Um, we need to be negative. One comma zero so here. Both of them are in W. As we know that zero times one is greater than or equal to zero. Um, I&#x27;m sorry. The definition here is you&#x27;re the vehicle zero. And we know that negative one times zero is greater than or equal to zero, since it&#x27;s just zero equals zero. Right? So when we add these two, we got, um, negative one comma one, which is not in W because negative one times one is not greater than or equal to zero because it&#x27;s negative one, which is actually less than zero. So it tells us that this ordered pair it would lie, Um, in the second or fourth quadrant. Um, therefore, it does not fit the definition of W, but you and me individually do fit the definition of W.

Samantha Lincroft · Answer

So if this problem were asked to show that you cross b is oh, if agonal to you plus fee and you minus C on. I like to think about this geometrically. So we have here to back dues you and the vector V and we know that by definition, the cross product you cross v is orthe agonal to you and is orthe agonal to be. And so because of that, we know that it&#x27;s orthe agonal to the plane containing U and V. So that means that as long as you plus V and you mind this fear in that plane, then we&#x27;re good and it is with agonal. And in fact, since we can define that plane as all of the points that can be reached by some linear combination of U and V, it must be orthogonal till that point. So the way that we can kind of think about this right is this factor here, for example, would be you plus B. And over here we would have you minus B. But all of these will still be in the same plane

Mike Gaerlan · Answer

So let&#x27;s start with the left side of the equation for the proof. So we&#x27;re gonna do what&#x27;s inside this print disease first. So we leave the on the outside and then multiply. See Time&#x27;s vite. So here we&#x27;re gonna have C V one plus C or sorry, comma C V two. And then when you multiply by the A, we&#x27;re gonna have a C V one comma, a comma, a C V two. And then, since a C is in both of these, we can factor out the A C as one constant. So that&#x27;s equal to the constant A C times V one comma V two or they see times the vector V. Now what this looks like geometrically is that if we had CV So let&#x27;s say this is CV here or this is the first. And then we do see V, and then we stretch that again by a here. This is a C V. Then if we were to do a C first, then, um, a c v, this one to be the same thing if we were just too, Um, multiply V by a C

Mike Gaerlan · Answer

to prove the equation. I&#x27;m going to start with the left hand side. So if you multiply a plus seat times we were gonna get a plus C times V one comma of an A plus C times V two. Then we can distribute the A, N. C and the V. So we&#x27;re going to get a V one plus C V one and then comma, and then we have a V two plus C V two. Now we can separate the, um some. So we&#x27;re gonna have a V one comma a V two plus and then c V one be one comma, see feed to then we can factor out the A and the C. So we have a, uh, looks a times B one comma V two plus C times V one. Me too. Or when you write this, we get a V plus C V. Now what that looks like geometrically is that if we were to have V and if we were to scale it by a plus C, let&#x27;s say is one and C is to So that means a plus c. It&#x27;s going to be three like that. Well, ah, a C is going to be this and then CV is going to be this. So when we add the two, we still get the same point or we still end up at the same point here.

Ankit Gupta · Answer

Prozac&#x27;s in off V Vector on crude A blue vector is equal Toe Director George W. Vector divided by more close or credible vector Police formally declared by the vector. If the projection off the vector on could&#x27;ve reflector V then week vector than we can say, that director is equal to director George W. Factor, divided by more glass or political director. Holy Square multiplied by the vector since V Vector daughter Double vector divided by more bliss or double vector Whole squad is scaler. Let&#x27;s assume that this is equal to okay. Then we can see that vector is equal. Okay, multiplied by W vector It means then one director can be expressed. Eddie scaler multiple Over there the vectors have the same direction. It means the projects on off Director onto Double Vector is director only. If director and the blue after have the same direction. It means any two vectors Director NW vector have the same direction, will satisfy the condition. Hope projection off The director wants Ow Vector is equal to the vector If v vector is equal toe k multiplied by W vector

Problem

Let $W$ be the union of the first and third quadr…

Problem 1 Easy Difficulty

Answer

a.) Yes. b. $c=-1$ and $u=\left[\begin{array}{l}{1} \\ {1}\end{array}\right]$

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

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Caleb E.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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a.) Yes. b. $c=-1$ and $u=\left[\begin{array}{l}{1} \\ {1}\end{array}\right]$

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Caleb E.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Anna Marie V.

Kayleah T.

Caleb E.

Joseph L.

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

Linear Algebra and Its Applications