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Question (40 points): Projection onto colA with ATA = [Cousider the matrixad the vector(see Unit 2, slicle 19) .a) Prove that 6 4 colA, that is that 6 cannot I expr...

Question

Question (40 points): Projection onto colA with ATA = [Cousider the matrixad the vector(see Unit 2, slicle 19) .a) Prove that 6 4 colA, that is that 6 cannot I expressed a5 linenr (Ohination of he cols of A, (Op) V) Check that / = prOf,A is ot hogonal to ech colum of (which iplies that 7is othogonal t0 col) ([Op) Cheek tat the SUI of the projeetions o 6 onto the direetions given by the colus of equal to its projection Onto COlA (20p)

Question (40 points): Projection onto colA with ATA = [ Cousider the matrix ad the vector (see Unit 2, slicle 19) . a) Prove that 6 4 colA, that is that 6 cannot I expressed a5 linenr (Ohination of he cols of A, (Op) V) Check that / = prOf,A is ot hogonal to ech colum of (which iplies that 7is othogonal t0 col) ([Op) Cheek tat the SUI of the projeetions o 6 onto the direetions given by the colus of equal to its projection Onto COlA (20p)



Answers

Decomposing a Vector into Components In Exercises $59-62,$ find the projection of $u$ onto $v .$ Then write $u$ as the sum of two orthogonal vectors, one of which is $\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}$.
$$\mathbf{u}=\langle- 3,-2\rangle$$
$$\mathbf{v}=\langle- 4,-1\rangle$$

Are so were given two vectors. The first fondness. Negative three. Negative too. The second long is negative floor negative form. All right, so now we want to find the projection. That's a step too long. Is too projection effect to you onto Vector V, which is given by the foreigner. You that prank with the over the enormous square temps Vector V. Uh, so let's do the math here. So here is gonna make it three times before 12 plus negative. Two times negative form. That's kind of it, too. And that the V Norm Square is gonna be a negative four square first native, one square. I wanted time 94 beautiful or so we do the math here is 14. Here it's, uh, 16 plus two. It's gonna be 18. So it's 4 to 6 16 plus one. It's gonna be 17. So it's gonna be 14 over 17 Negative. Four negative long. So we want to plug everything into this. Break it. So it's gonna be 56 negative 56 over 16 and maybe 14 over 16. All right, so that's the projection off you on to be. So now we want Thio friend, the authority. No, Lecter just stopped it too, Which is given by you mind stuff you want. It's gonna be a load of three. Negative, too. Minus negative. 56. 16. Negative. 14. 17 to these 17. So the answer is five 16 native. 20 17. All right, so now we can ride you in terms of two terms, which are authority. No. So the 1st 1 is Stop you want? Which is negative. 56 over 16. Nearly 15 over 16. 17 plus fine. Over 17. Note of 20 over 17. Right. So this two of the answers

All right. So the question is to find the projection affect her You onto the vector v. So Dr Use Given by Ford too. And vector V is given by long negative too. All right. We can just used a full mirror where the projection it's gonna be, Well, the projection of the you on to re it's gonna be duck product off this too. Oh, the normal square times the vector V. So if we do the math, it's gonna be four times One is four clothes, two times negative to is negative for. So that's the top product over V Square VI Enormous square says it's gonna be one. We're was negative. Two sweeter times. One negative too. All right, because we have here the numerator being zero, So the answer is gonna be a zero and zero, So that's well, give me a w walk R a W two. It's gonna be you minus stuff you want. We're just gonna be you, right? Because w one is just 00 or so. We want to write the you as the sum of two vectors. So you No, it's gonna be for two, because zero zero. All right, So now we know that the projection of few on V it's gonna be 00 And if you're gonna write the view on two vectors, it's gonna be 42 plus 00

They're so in this occasion we're going to consider the space of polynomial of degree two. With the revelation in their product. That means that we have a set of points where we are going to ever eat our point annuals. And what we're going to do is taking the sum of the multiplication of these two point almost evaluated at those points. So in this case the points that we are going to have a wider point annuals are zero equals 2 -2. The zero other point to. And these are the point of us that we have here. So what we need to show is that these two point terminals are orthogonal, which is equivalent that the inner product Is equal to zero. So let's take these let's calculate this uh in the product. So then they're proud of B Q evaluated at the point x zero x one x two. It's going to be equal to be x zero cube exit europe plus P x one Q x one loss P x two, cute X two. Then Let's avoid the points. So we got here. Extras -2. So be evaluated at -2 is just -2 times minus two square for the point x one. That is equal to zero. Both point animals will be equal to zero, so it's just 10 And the last term is To hear so it's two times 2 square. You can observe that this is equal to minus eight plus eight, which clearly is equal to zero. Therefore these two point normals are orthogonal.

Given Victoza, you're is a cradle We're going like and movie is six comma one No, the projection off you on the Lucas name it w one will be equal to on the dot product. Don't you envy? Divided by the square off the magnitude of the multiplied by the rector We So this is a skater Multiplication off the vector v with this scaler dome. So now the doctor you envy is a good time. Six. That is well plus brooding one That is point no, The absolute value or the magnitude of movie is 36 plus one That is great. Route off 37 then we have a whole square. So this will be only 37 you know, multiplication will be written six common one. So this becomes 84 upon 37 comma 14 upon the zone. This is the projection off you only which is W one. So this is one of the toe component we want to result doing toe or the one of components the other woman notice college w toe will be given by U minus. Got you one minus 84 upon 37 coma for Dana Point, the seven and on doing the subtraction. We even get minus. Been upon 37. 60 on 37. Does we can write you, uh, some off two components, That is. First we have w one 84 on 37 Omar 49 37 plus the second component of due to richest minus stand on our decision. Coma 60 on 87. So this is you resulting toe to complain.


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