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17 . Consider the mapping L : Pz ~ Ps defined byL(ao + a1t + 02l2) = dot + 012 + 02t8 2 3 Show that L is a linear transformation....

Question

17 . Consider the mapping L : Pz ~ Ps defined byL(ao + a1t + 02l2) = dot + 012 + 02t8 2 3 Show that L is a linear transformation.

17 . Consider the mapping L : Pz ~ Ps defined by L(ao + a1t + 02l2) = dot + 012 + 02t8 2 3 Show that L is a linear transformation.



Answers

Given $\mathbf{u} \neq \mathbf{0}$ in $\mathbb{R}^{n},$ let $L=$ Span $\{\mathbf{u}\} .$ Show that the mapping $\mathbf{x} \mapsto \operatorname{proj}_{L} \mathbf{x}$ is a linear transformation.

I was 25. 30 of zero is zero safety off zero Vector by definition, is a cordial zero comments. The zero is the zero vector So far, it's probably service would notice, Then verify that it respects tradition. So let you and Reba to Victor's with coordinates. E When you do you 300 103. So? So. This by definition, is a cordial. You want less? We want less. Three times. You're hopeless. We do this V three plus years. Three. And this is a quartile. And the second coordinators acquittal. You want press? We want minus your doctors with. This is a little You want those three or two close? You three going while you one minus You too. Knows the one? No Street veto. Yes, really. City. This is you, Terry. Comb of even minus we toe. This is 1/4 tee off. You one, you do. You three lusty off the one with the three. Let us know. Check that. It respects scale a multiplication. So, Theo, see names you want. When will see turns you toe common See times you three Is it called? See you. Those tea names. You'll go. Let's see you three. Come and see. You want my new See you. See you one minus you. This is a call to see times the victor we want plus three with toe so that this is you want you want you want us through You toe this you three go my you on minus you. And this is a call to see times the vector the you want you do you three Therefore, this might have given to us easily in a transmission.

In this example, we're starting off with a function f mapping from our end into RM and this particular transformation is given to be linear. The next assumption we have is that a set s consisting of three different points coming from our end is finally dependent. Using these two statements, let's see what we can determine about the set of images F of s which is equal to F A p one f of p two and f of p three. Let's start then with the statement that s is finally dependent weaken, say bit by the statement here that there exists scale er's let's call them see one c two C three and these are not all zero and these satisfy the following. So it's say such that if we formed the combination, see one p one plus C to p two plus C three p three, and consider this as a vector equation so equal to zero. Then we have that at the same time. C one plus C two plus C three sums to zero. This is our full definition of what it means for the vectors. P one through p three dippy at flying F finally dependent. Next, let's go back to the transfer formation F where we know it's linear. If we go to our vector equation, we can evaluate both sides with the function f so we'll have f of c one p one plus C two times p two plus C, three times p three and this is now equal to F of zero vector in our end. Now we know a couple handy tricks here whenever we're dealing with a linear transformation. First, the right hand side is automatically does your vector in our M this time, since leaner transformations take the zero vector back to the zero Vector, we also know that on the left hand side we conform the following some take c one times f of p one plus C, two times f at P two plus C three. Let's go with green for C three times f at p three. So that's what linearity provides us in this situation. But now it's noticed. The left hand side is interesting. We have scale er's C one, C two, C three and they still all some 20 There involving the vectors F of p one f of p two f of P three and it's set equal to the zero vector. This tells us immediately. By the definition of after independence, the FFS must be at flying Lee dependent. So it's right that down as our conclusion this shows f of s is finally dependent. Since the same scale er's see one suits through C three gives us the dependent relation where the scale er's some 20 So what this problem is telling us all together is, if we have a transformation that's linear, it will take a set that's already a finely dependent and map it into a another F finely dependent set each and every time.

All right. So for this question, you want to find out if the linear transformation of the norm is going to be um a if the transformation of uh of a vector from our three to the norm is going to be a linear transformation. And we're going to look at homogeneity. And we're going to look at the second condition which is going to what you're going to compare the transformation of two vectors. The addition of two vectors. Yeah. Uh then transformed compared to to transform vectors, then added, I'm gonna be the same factor. So 101 and 010 Mhm. So for the first, for the first time you can see 10 one plus 010 is just gonna be 111 And the norm of that is just going to be all of the each individual vector squared and then added and square root. So that's going to be the root of three because one square dishes one. And then when we transform these vectors first into the norm, we then get one squared plus zero plus one. So it's going to be a roof two plus zero plus one squared plus one, which is gonna be one. Just gonna give us the root of two plus one. And since these are not the same thing, I mean they're not equal. And so this is not a linear transformation. Mm. Yeah.

Transport off zero matrix zero metrics and you feel at zero No zero, you get zero. Therefore sa 00 Your district, um, interests is MB and find out what as off a plus we is. This is a corridor unless we have strong support off fitness week. No transpose off some off. The matrices is trance falls off. If let's be transfers is equal to a trance was be transfers. So this is equal to a place transports and make Extradition is going maturity. So we get that this is 1/4 Plessy transport, less weakness. We transfers. This is 1/4 less Sophie, and this is 1/4 less off me. So we have That is respect. Tradition No s off, See, Times of matrix is a quarto. See time say, let's see times a glance Waas no, From the property of transpose transposes transfers of seed names, ese courtesy names a trance pools which is a courtesy off a plus transpose which is a little See your names. That's all. Therefore s is a linear transform


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