Were given a subspace W. Which is the set of all polynomial of degree at most. two P. Two with an inner product inner product of F. And G is the integral from 0 to 1 of fft times. G M T E T. So oh yeah, interested in the projection of the function F f T equals T cute onto the space. W. As a hint, we're told you the orthogonal pollen or meals one To T -1 -1 and 60 squared minus 60 plus one. And then you can So we found that these were orthogonal in exercise 722. Yeah. Now because these are our dog kennel, we can just calculate the fourier coefficients to find a projection of that's under the space. Yeah, these are an orthogonal basis. Mm mm for W. Okay. Yeah. So before a coefficient C one is in a product of t cubed with one Over the inner product of one with itself and Yes. Right. Yeah. So this is the integral from 0 to 1 of t cubed D. T Over the integral from 0 to 1 of one gt. This is uh 1/4 Over one which is 1/4. Yeah the second fourier coefficients. C two is the inner product of t cubed with two t minus one Over the inner product of two T -1 with itself. This is the integral from 0 to 1 of well, t cubed times to t minus one DT over. The integral from 0-1 of to t minus one squared E T. This is equal to the integral from 0 to 1 of two T to the fourth minus t cube pt over the integral from 0 to 1 of four, T squared plus minus 40 plus one B two. Right. And this is equal to Take me up to derivatives and evaluating to 5th drugs. 14 over four thirds minus two plus one. This is to Fix. -14 is 8 20th -5, 20 assists 3 20th We were 4/3 -2 4/3 6/3 is negative. 2/3 plus one plus three thirds is positive. One third which is mm 9/20 right. You know, I like like in the field and finally for a coefficient C3 this is the inner product of she accused, I will be with uh then, I mean New Orleans, these Chicago, the 21st and 22nd, 60 squared -60 plus one. Yeah, pleased by the over the inner product of 60 squared minus 60 plus one with itself. What he said This is the integral from 0 to 1 of t cubed times 60 squared minus 60 plus one. Bt over the integral from 0 to 1 of 60 squared -60 plus one squared E. T. Yeah back I mean mhm. The this is the integral from 0 to 1 of uh 60 to the fifth minus 62. The fourth plus T cubed E. T. over. The integral from 0 to 1 of this is a little tougher. Doctor said 36 T. to the 4th. Mhm. 36 T. to the 4th -72 T Cubed. I do act plus 48 T squared minus 12 T. Mhm. No it's somebody that cares. Plus one. Well he's not taking anti derivative and evaluating we get one minus 6/5 plus 1/4. Yeah. Yeah. Mhm. 36 5th minus. Yeah. Yes. Thinking of buddy 18. Mhm. Yeah. Plus uh 16 minus six plus one. This is uh 40th minus 24/20 is negative. 4 20th Plus 5 20th says positive. 1 20 over. Uh six in Melbourne, Melbourne that 36/5 95th. Yeah, This is 1 5th he looked Which is 5 20th so or 1/4. Mhm. And therefore, oh okay, projection of our function F onto our space W This is going to be C1 times are first function one Plus C, two times their second function to T -1 Plus C. three times our third function 60 squared minus 60 plus one. So, plugging in this is uh 1/4 plus 9/20 Times to T -1 plus 1/4 times 60 squared minus 60 plus one. Mhm. This sympathize too. Three halves, t squared minus 3/5 T plus 1/20.