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Let $V$ be the space $C[-2,2]$ with the inner product of Example $7 .$ Find an orthogonal basis for the subspace spanned by the polynomials $1, t,$ and $t^{2} .$

$\left\{1, t, 3 t^{2}-4\right\}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 7

Inner Product Spaces

Vectors

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we have the polynomial speed zero p one MP too, and they don't know the constant function. One the degree. One problem lt in the degree to play multi square. We're looking at this bully normals as vectors in the vector space off continues function from minus two, two plus two with dinner problem given by the integral off the product. So we're we're looking for are Fogell basis for the span off zero p one MP too. So we're going to use the gram Schmidt the composition method. So let's begin by defining a vector zero off the so this will be the first ah, element off the air for colonel basis. And let's just put it equal to zero off team thought is the constant functional one. Let's also compute dinner problem at zero against zero just because it will need later. So it is by definition of inner product, the integral from minus 2 to 2 off well at zero times of zero, so one times 11 So just in there of Juan de ti aunties, just uneasy computation is is for now. Let's project the one until at zero. I remember that this is given by P one against a zero dividing at zero against zero, and this number multiplies the vector of zero of tea. Now I put one over four and that's the ah in their product at zero against zero in the denominator. And now we have in the numerator p one against zero. So that's the entire from minus 2 to 2 off just x the X net. I'm using X because I'm using t in another place. But remember, it is just Ah, dummy. Variable token is called the axe, and this number multiplies the function of zero T. But now X is a not function and we're integrating over somatic domain. So disintegrate zero. And so these old projection is zero. Now we can proceed with the good screen, the composition. So we put F one off teen as everyone off the minus the projection off the one onto a zero. But now we've seen that the projection is just zero. So let's just be one of tea. So the function t And again, let's compute f one against a form because we need you later. Disease didn't take it out from minus two 22 off the phone skirt, Softy square the tea and again, easy computation. 16 over three. And now, finally, that's computer projection onto F zero off B two off t. So similarly, this is B two against zero. Dividing have zero against 00 f. Zero f t and disease 1/4 date. A girl minus 2 to 2 off X where the ex times have zero off t that did this 4/3. They're multiplies at zero of tea, but it's just because of function one. And now we complete the projection on life. One off, Petey off. T Saudis be two against one dividing F one against one. These old number that multiplies the vector F 1 50 disease three over 16 Didn't a girl from minus 2 to 2 off B to F one. So that's X cubed Jax. Everything multiplying the vector of fun of tea. But again, excuse. He's on all the function on automatic domain. So these old entirely zero and therefore the projection is zero. And now we can complete the Gram Schmidt the composition because we put half to off t to be pizza off t minus the projection onto a zero off. He's off t minus the projection onto F one off the tee off T and Disease T squared minus 4/3 nine minus zero. So the vectors f zero f one.

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