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2/s118anikaikhtm2360sD01/Hws/20/2effectivelUser-jaciemit&key-sDNonKAvor9mukMh; MATHEMATICAL ASSOCIATION OF AMERICA Loga webwork/ $118anikakhtm2360sd01/hw5/20 HW5: Problem 20 Previous Problem List Next (1 point) Use the Gram-Schmidt orthonormalization process to transform the given basis for Rt into an orthonormal basis Use he Eucidean inner product for Rt and use the vectors in the order in which they are given Answer Preview My Answers Subm Answers You have attempted this problem O times You have unimited attempts remaining

Explanation / Answer

Let v1 = (3,4,0,0), v2 = (-1,1,0,0), v3 =(2,1,0,-1) and v4 = (0,1,1,0).

Also, let u1 = v1 =(3,4,0,0), u2 = v2-proju1 (v2) = v2-[(v2.u1)/(u1.u1)]u1 = v2-[((-3+4+0+0)/(9+16+0+0)] u1 = (-1,1,0,0)-(1/25)(3,4,0,0)=(-28/25,21/25,0,0),u3=v3-proju1(v3)-proju2(v3)=v3-[(v3.u1)/(u1.u1)]u1-[(v3.u2)/(u2.u2)]u2= v3-[(6+4+0+0)/(9+16+0+0)]u1-[(-56/25+21/25+0+0)/(784/625+441/625+0+0)]u2 = (2,1,0,-1)-(10/25)(3,4,0,0)+(35/25)*(625/1225)(-28/25,21/25,0,0)= (2,1,0,-1)-(30/25,40/25,0,0)+ (5/7)(-28/25,21/25,0,0) = (0,0,0,-1) and u4=v4-proju1(v4)-proju2(v4)-proju3(v4) = v4-[(v4.u1)/(u1.u1)]u1-[(v4.u2)/(u2.u2)]u2-[(v4.u3)/(u3.u3)]u3 = v4-[(0+4+0+0)/(9+16+0+0)]u1-[(0+21/25+0+0)/ (784/625+441/625+0+0)]u2–[(0+0+0+0)/(0+0+0+1)]u3= (0,1,1,0)-(4/25)(3,4,0,0)-(21/25)*                       (625/1225)(-28/25,21/25,0,0)= (0,1,1,0)-(12/25,16/25,0,0)-(3/7)(-28/25,21/25,0,0)=(0,0,1,0). Then {u1,u2,u3,u4} is an orthogonal basis for R4. Now, we have to convert u1,u2,u3,u4 to unit vectors. It may be observed that u3 = (0,0,0,-1) and u4=(0,0,1,0) are already unit vector. Let e1 = u1/||u1|| = (3,4,0,0)/(16+25+0+0)1/2= (3/5,4/5,0,0), e2 = u2/||u2||=(-28/25,21/25,0,0)/( 784/625+441/625+0+0)1/2 = (-4/5,3/5,0,0), e3 = u3 = (0,0,0,-1) and e4 =u4=(0,0,1,0) . Then, {e1,e2,e3,e4} ={(3/5,4/5,0,0),(-4/5,3/5,0,0), (0,0,0,-1), (0,0,1,0)} is an orthonormal basis for R4.    

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