Question

Find the orthogonal projection of v=[−2,10,−16,−19] onto the subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

Find the orthogonal projection of v=[−2,10,−16,−19] onto the subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

Homework Answers

Answer #1

Let v1 = [-4,0,-2,1], v2 = [-4,-2,5,1] and v3 = [3,-1,-3,4].

We have projv1(v)=[(v.v1)/(v1.v1)]v1=[( 8+0+32-19)/(16+0+4+1)]v1=v1=[-4,0,-2,1], projv2(v)=[(v.v2)/(v2.v2)]v2=[( 8-20-80-19)/(16+4+25+1)]v2 = (111/46)[-4,-2,5,1]= [-222/23, -111/23,555/46,111/46] and projv3(v)=[(v.v3)/(v3.v3)]v3=[(-6-10+48+76)/(9+1+9+16)]v3 = (108/35) [3,-1,-3,4] = [ 324/35,-108/35,-324/35,432/35].

Hence, projW(v)= projv1(v)=+ projv2(v)+ projv3(v)= [-4,0,-2,1]+ [-222/23, -111/23,555/46,111/46]+ [ 324/35,-108/35,-324/35,432/35] = [-3538/805, -6369/805,1301/1610,25367/1610]

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