Please answer each question with complete work. Do not have wrong answers, pleas

Question

Please answer each question with complete work. Do not have wrong answers, please.

Consider the two vectors u = (1,0, - 1), v = (1,1,1) and the inner product in R3 given by: (u,v) = ujri + 2u2V2 + u3v3 compute the following: ||u||, 1|v|| and ||u - v||, the norms induced by the given inner product. The angle 9 between u and v. projv u and projuv Let f(t) = t2 and g(t) = e -t and the inner product in C[-1,1] given by: (f, g) = dt/ compute the following: ||f||, ||g|| and ||f-g||, the norms as induced by the given inner product. The angle theta between f and g. projf g and projg f Let W be the subspace of R4 spanned by the vectors {{3,4,0,0) , (-1,1,0,0), (2,1,0, -1)}. Use the Gram-Schmidt orthonormalization process and standard inner product (dot product) to find an orthonormal basis for W and Find projwu, the projection of the vector u = (0, 1, 1, 0) onto the subspace W.

Explanation / Answer

5)

a)

||u|| = sqrt <u,u> = sqrt ( 1*1+2*0*0+-1*-1) = sqrt(2)

||v|| = sqrt <v,v> = sqrt ( 1*1+2*1*1+1*1) = sqrt(4)=2

u-v = (0,-1,-2), so ||u-v|| = sqrt(0+2*-1*-1+-2*-2)=sqrt(6)

b) cos(theta) = <u,v>/(||u||||v||) = 0, so u and v are perpendicular, theta = Pi/2

c)

Since u and v are orthogonal :

proj_v(u) = <v,u>/<v,v> v = (0,0,0)

proj_u(v) = <u,v>/<u,u> u = (0,0,0)

6)

a)

||f|| = sqrt <f,f> = sqrt (int(-1<t<1) t^4 =sqrt( [t^5/5] ) = sqrt(2/5)

||g||= sqrt <g,g> = sqrt(int(-1<t<1) e^(-2t) = sqrt(1/2(e^2-e^(-2)))=sqrt(sinh(2)) (hyperpolyic sinus)

We can use : ||f-g|| = sqrt ( ||f||^2-2<f,g>+||g||^2 ) or do all the computation by hand

<f,g> = int t^2 e^(-t) = -t^2e^(-t) + int 2t e^(-t) = [-t^2e^(-t) -2te^(-t) -2e^(-t)](-1<t<1) using integration by part

<f,g> = e-5e^(-1)

So ||f-g|| = sqrt ( 2/5+sinh(2) -2 (e-5e^(-1))

b) cos(theta) = <f,g>/||f|||g|| = sqrt ( 2/5+sinh(2) -2 (e-5e^(-1)) / (sqrt(2/5)sqrt(sinh(2)) ) = 1.25

so theta ~ acos(1.25) ~ 0.69 radians (39.75

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