the-figure-shows-vectors-mathbfu-mathbfv-and-mathbfw-along-with-the-images-tmathbfu-and-tmathbfv-und

Question

The figure shows vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w},$ along with the images $T(\mathbf{u})$ and $T(\mathbf{v})$ under the action of a linear transformation $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} .$ Copy this figure carefully, and draw the image $T(\mathbf{w})$ as accurately as possible. [Hint: First, write $\mathbf{w}$ as a linear combination of $\mathbf{u}$ and $\mathbf{v} . ]$

Ashley Boni · Accepted Answer

So we&#x27;re given these two figures and we want to try to plot t of w So the hit tells us first right? W as a linear combination of U N v. Eso First, let&#x27;s look at this picture on the left. Let&#x27;s connect V plus venue or W and you And we know this vector is parallel to V. So let&#x27;s try to use the parallelogram room. Um, so let&#x27;s slide this vector downwards along you. Um, until the, um, the back point here is that the origin? So we do. So we get this vector here and we could see the length of this is about twice the length of the So all I&#x27;ve done is taken this red vector, slide it down. And since we crossed the origin of points in the other direction So this red factor is to V about, um So let&#x27;s go ahead and connect this. So if we connect from, I had to tell this vector since, um, the specter here and this vector here are parallel in the same length. We know that this is also you. These two are the same. Ah, so now we have a parallelogram here, so this tells us that w is equal to you. Plus two b So now we want to find t of w So let&#x27;s just apply t to both sides, So t of w as he called a t of you plus two b And so we&#x27;re now we&#x27;re gonna use the rules of linearity. So this is the same ass t of you plus to t of the cause. Linearity tells us that we could pull constant out. So if t of w is equal to t of you, which is this vector here, plus twice t of me first, let&#x27;s find Let&#x27;s do two TV then. So that&#x27;s gonna be double this length. So this red rector is too t v anti of you is this one. So let&#x27;s go ahead and connect those two things. Ah, First, actually, let&#x27;s charlety of you up here. We&#x27;re just gonna slide it up because we want to make sure these air connected And now well, you will just, uh, connect from the origin. So let&#x27;s do that. Another color. So here&#x27;s our t of W. Because we have to make sure the reason we couldn&#x27;t have just connected used to is because we&#x27;d be correcting the, uh to head. So we have to have everything had detail, so we needed to shift it up.

Aparna Shakti · Answer

in this problem. It is asked to copy the vectors in the figure and use them to draw the following vectors. So here are the vectors given to us in this way. Right, So these are the three factors. Alright, So here you is here, right? We have made here. Right. And finally we are having this uh we and uselessly. Right. So we have just taken this mistakes like vectors. Okay, And just we are putting in this way. Right? So we know that a vector has been following the uh step, right, this is in you then. We this has some direction. Right? This is some direction. This has some directions, right? And the final direction will be opposed to it. Right? This is opposite, right. This is not going in this direction, right, It is going in this direction. Okay, so this is this is the common observation, we we just observe and the factors. Right? So here we can say uh here. Now for the second part we can see we are asked to find you please. W so this is also very simple, we will put just you and we allow uh sorry, you and w with it. Right, So this is also following uh direction. Right? And the opposite A deposit direction will be having from initial point to the final point. Right, So this was an initial point in the previous problem. Right, And this is our final point. So the direction will be opposite, remember. Right, so this is your place. W we have got All right, so moving ahead we will see for we plus W Right, so now here we we is given to us. Where is we? Yes, this is Director. Right? We and we can draw it as it is and you place W. Right, so w is in downward, so we will just uh yeah, we will just take a tail. You can say this is I suppose this is a fact. Right? Let us see. So the sustain and this is alright, this should be head. I&#x27;m sorry, they should be head and this is jane. Yeah. Right, so the uh you can say this, this becomes very easy. So I&#x27;ll tell you. So this headed of this head is towards upright for me. So now, so from the head of the v detail of the W should start. Right? So this will go in this way. So this is the direction. Right? And the final direction will be opposed to both of them. So we can see this is this is our presentation. We are getting people&#x27;s W here as a resultant factor. All right, so now moving ahead, we want to calculate the humanist humanness. We So this is also very simple, we isn&#x27;t it wasn&#x27;t a proper direction and we know that in the vectors if we if we draw or if we change the direction opposite to it. Right? So it becomes minus, we sign also changes. Right? So this becomes minus we and this is how we make the uh direction, party, final resultant vector. Right? So this is a final resented resulting back that. Right, okay, so now moving ahead we want to calculate for you plus V plus w. Right, so we have drawn you plus Vdo and we have drawn here on W. Right, so this should this will not make any triangle, right? Because there are three vectors to be, it is making up quadri later. So this was the initial point, let us say you plus weekend Britain as we plus you? Yes, so yes, we have a common value here and the destination point was this? So this is a final vector, this is useless, we blister Bolivia. Okay, so this is very simple, this is you place me right, this is headed headed towards this, uh the head or head of the, you can see you write and the tail of free, you can say right, okay, Mhm. So now moving forward, we are asked to find out u minus b minus u minus the blue minus v here, So this is for sure, there will be four points here. Right? So here it will be you and -5 is nothing. But this is drawn opposite to the given direction, it was downwards in the given direction if we just inverted. Right? So uh we can make it on the upper up direction and it will be minus W. Right, similarly for minus, we we have already discussed above, so this will be minus, we write and the resultant factor will be From initial .2 final point. Right, so what is our initial point? Initial point is nothing but the tail of the tale of the you went to and head of the you can see a head of the minus move back. So this is how we calculate the u minus U minus w minus week. So this is how we calculate this vector. I hope you understand the concept. Thank you for watching.

Shamshad Waris · Answer

here, Kitty. Viktors, Viktor, You make in the end, will grow the victor. You polis! You blessed here have to get her off. You bless me. So fasting the better. You better you eat you so, uh no. Better be so better. Be mean the letter B and place it stay. So still is this one the head off? Hector B and the head off The place is still in the place, up in the head. They had a better you is here so weekend. But let here and it still is less And the head off today This is the result of the vector. You and he independent like that. So you bless. Be using that. Bring us for about keeping a lower orbit. No, we can&#x27;t depend chicken once. So you give the one needs. You bless. Dumb question will grow you better. So you get these this way Now plus letter b So we don&#x27;t let the victor double and place it still less it been. And the Hidalgo picked up better so that they hit it off. The better you. This was a tale of the vector. W it here. So at let here. So you get like that? Is that extending the gelatin of that? This both victories You did not bring those law? Is this one? So the better pupil us doctor Doesn&#x27;t no it draw next one Sure card people s so past winter. All the nectar the bees made that the letter B No better. So we just let the victor dumb and place It&#x27;s still at a head off. Better be so that fill up they blew it. This is why a place at the head, the head ups or the head of the Victor Beach. This one So that let here the news you get no Look at the Children of that thesis Victories you didn&#x27;t being us law, he diesel. So this one is that people s double now similarly will drop. They picked up people as you Pulis Double first under other better be so the better bees. MySpace. So I think this one now detector double letter you So we translate the victor. You extend and place it still at the head of the victor Be Then he will get here with Victor. You know better Double. So we just let the rector double and less. It&#x27;s still at the head off the victor b Make that You didn&#x27;t get like that, Victor. I think that this is starting to gel. So this rectory you pull s people that you blessed No, we can build the victor they&#x27;re missing. This is the the party, not a party age. You might not be possible. There are the better you here. So the fact that you eat the soil and let there be so the better beat letter B may be but we haven&#x27;t given a minus vector. So that mine Inspector, the authorities have the better visited the open it up. So they know that I will do it so that this is on this victories. You, my necessarily similarly really grow Last one Last one Is he Do you u minus w minus b So pastoral There are the victor You. So the fact that you it This one and the victor double suspected Bablu each usually you Mrs, But I have the better w minus No bees. So the victim be We just let the letter b and place it still at the metadata Better. My so they said be a This is what this is B but we have it given an minus effectively So taken up with a better be the minus extent we know the result conduct that this big studies You and the kid tell you that still do this better is called. Do you mind if w minus b? Thank you.

Michael Jacobsen · Answer

here. We&#x27;re going to be considering an example that has a lot of moving parts. First we&#x27;re doing dealing with the transformation T. Which is mapping V two w. But the most important thing about this transformation is that it is linear. The next big piece of this puzzle is that we have a subspace you of the vector Space V, which is code comes from the domain of tea. So we have a subspace moving around and the second moving piece of this puzzle is this space t of you. This is contained in the CO domain W And here is our goal. We want to prove that this is, in fact, a sub space of w not just a subset. Well, here&#x27;s how we&#x27;re going to start. If we&#x27;re going to prove that this is inside W we should say exactly what t of you is that sets our notation and it helps us move through the solution. So we have by definition, that t of the subspace you is going to be equal to the set of images which are denoted by t of X, such that ex itself is in. You so t of you here It is a set of the images t of X which we&#x27;ve indicated. And the only requirement is that the vector X had to have come from you, which is a subspace of V. This takes a moment to digest, so there&#x27;s no harm to take a few moments Drawing pictures are looking at diagrams now, since we&#x27;re doing a subspace proof are proof comes in three parts And the first part is to show that the zero vector is in t of you. Well, to start this one, we can say Since t is linear, this zero vector we&#x27;re looking at can also be expressed as t evaluate at the zero Vector. Since linearity allows us to take tea mapping is your vector always back to the zero vector but notice what we&#x27;ve shown So far the zero vector has been written as t apply to the zero Vector. So this is your vector here is playing the role of X and it is definitely in you because a subspace you always contains its zero vector. So we can say that this implies that zero vector is in t of you. Since the zero Vector here is in you So let&#x27;s go off to the second part. For the second part, we need to grab to arbitrary elements of t of you. So let&#x27;s say let t of x and t of. Why be in t of you? It&#x27;s sometimes helpful to write out what our goal is. We need to show that t of x plus t of y is also in t of you. So let&#x27;s start off by writing down TF x plus t of y and see what we can do from here. Well, the first thing we could say is because the mapping Trenti is linear. This is the same as T of X plus y. And that means we have tea of a vector X and X plus y is in you because you is a subspace, the linear combinations x and y. If we take X plus, why we&#x27;re still in the subspace you now we know x and why came from you? Because that&#x27;s how we define the set t of you itself. So since we have and X plus y here matching this X where X plus y is a new it follows then or imply it&#x27;s implied that t of X plus t of y is in t of you notice that the only way that this conclusion here is sound is if we use the fact that T is both linear and you is a subspace of V at the same time for part two. Now let&#x27;s go to Part three in part three, work still going to use the T of X. That&#x27;s provided, but we also say, Let see be in our. Then again, it&#x27;s important to stay our goal either mentally or just writing it down. And our goal is to show, or we need to show that C Times T of X is in t of you. Let&#x27;s see how to show that. First, I&#x27;ll start with C times t of X, the vectoring question and now we use linearity again. This is the same as tee times. See Adex And then at this stage, we use the fact that you was a subspace of V. That means since X came from you see, Time&#x27;s X is still in you. So this implies that C Times t of X is in t of you since see Time&#x27;s X is in you. So we&#x27;re able to now verify parts 12 and three for the definition of subspace. And so our conclusion is the following t of you is a vector subspace of w. So what we have shown altogether is if we have a subspace you than the set of all images is also a subspace of the co domain.

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna be solving problems over 26 of section 2.3, which is based on linear independence. Ah, transformation or not, linear independence. I&#x27;m sorry. It&#x27;s based on determinants and Cramer&#x27;s rule and, uh, linear transformations and volumes. So here we&#x27;re given that Tia&#x27;s transformation, um from our himto are in and we know that p is a vector and s is a set in r r m s as is a set in, um our, um So let we can let this is a proof so we can let the vector v be any vector any vector from the set s which is on our M. And by definition, we know that people s is a set of all vectors because s a set of all vectors, um, better of the form p plus V that we did find your love. Uh and we know that V isn&#x27;t s so applying t to ah, typical vector and people cess. We have, um t and we apply people. Savi innit would give us tp plus t v. And this is this vector is in the said denoted by t plus s her teepee plus t s because we know that he is then s This proves that t maps the set people says onto the set T p plus t s. So this proves that, um, t as t maps people us us onto t p plus t s was what the initial problems asking which stated that we need to show that the image of people says under T a translated set t put people&#x27;s t s in rn. So there&#x27;s just a simple proof so shit.

Eric Wendland · Answer

So we have our grid here and let&#x27;s say we want to find factor VII onto So let&#x27;s say we have factor w going out like this and we have you&#x27;re they going out? So, quite simply, what we&#x27;re gonna end up doing to be alone. We&#x27;re gonna take this adjustments here. W what we&#x27;re gonna d&#x27;oh at this point right here. End of vector V and we&#x27;re gonna project it down. So is our Sahgal vector w right here. This is basically what you&#x27;re gonna do, what you&#x27;re gonna find the projection of beyond to w gonna got to the end of the and basically draw line that is perpendicular right here with W.

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

Linear Algebra and Its Applications

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Maria G.

Kristen K.

Absolute Value - Example 1

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Linear Algebra and Its Applications

Anna Marie V.

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Absolute Value - Example 1

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

Anna Marie V.

Heather Z.

Maria G.

Kristen K.

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

Linear Algebra and Its Applications