Math Problem Linear Algebra Assignment Paper #4 - 550 Words

Math Problem: Linear Algebra Assignment Paper #4 (Math Problem Sample) Content: Linear Algebra: Assignment 4University AffiliationStudent NameCourse NameDateIntroductionAssignment 4 of the linear algebra module deals with elementary vector operations with applications of algebra methods to solve linear systems of equations. Methods, properties and theorems are adopted from Howard Anton Chris Rorres book, Elementary Linear Algebra (11th Edition). The solutions are herein indicated stepwise and in order of questions as they appear in the original assignment file.Question 1(a)u=(1,2,-3,5,0)v=(0,4,-1,1,2)w=(7,1,-4,-2,3) * v+w=0,4,-1,1,2+7,1,-4,-2,3=0+7,4+,1,-1-4,1-2,2+3=7,5,-5,-1.5 * 3(2u-v=3(21,2,-3,5,0-(0,4,-1,1,2))=3(2,4,-6,10,0-(0,4,-1,1,2))=3(2-0,4-4,-6+1,10-1,0-2)=3(2,0,-5,9,-2))=(6,0,-15,27,-6) * 3u-v)-(2u+4w)==31,2,-3,5,0-0,4,-1,1,2-21,2,-3,5,0+7,1,-4,-2,3=(3,6,-9,15,0-(0,4,-1,1,2))-(2,4,-6,10,0+(28,4,-16,-8,12))=3-0,6-4,-9+1,15-1,0-2-2+28,4+4,-6-16,10-8,0+12=3,2,-8,14,-2-30,8,-22,2,12=(-28,-6,-30,12,-14) * 12(w-5v+2u)=127,1,-4,-2,3-50,4,-1, 1,2+21,2,-3,5,0+0,4,-1,1,2=127,1,-4,-2,3-0,20, -5,5,10+2,4,-6,10,0+(0,4-1,1,2)=127-0+2,1-20+4,-4+5-6,-2-5+10,3-10+0+0,4,-1,1,12=129,-15,-5,3-7+(0,4,-1,1,2)=92,-152,-52,32,-72+0,4-1,1,2=(92,-72,-72,52,-32)Question 1(b)Given:c1-1,0,2+c12,2,-2+c31,-2,1=(-6,12,4)-c1+2c2+c3=-6 i2c2-2c3=12 ii2c1-2c2+c3=4 (iii)Solve eqn i and eqn (iii) to eliminate the variable c1-2c1+4c2+2c3=-122c1-2c2+c3=4Adding the two equations, the following resultant equation is obtained:2c2+3c3=-8 (iv)Solving equation (ii) and equation (iv):2c2-2c3=122c2+3c3=-85c3=-20c3=-4 c2-c3=6 c2=2Also:c1=6+2c2+c3c1=6+4-4c1=6Question 2(a) * Given:u=(1,2-3,0)v=5,1,2,-2d=1-52+2-12+-3-22+(0--21)2=16+1+25+446andcos=u.vuv=1*5+2*1+-3*2+0*212+22+(-3)2+02*52+12+22+(-2)2=5+2-6+01434=11434=cos-1(11434)=87.37Hence 900 * u=(0,1,1,1,2)v=2,1,0,-1,3d=4+0+1+4+1=10cos=0+1+0-1+61+1+1+44+1+1+9=6715=cos-1 67=54.150 is an acute angle since 900Question 2(b)According to Cauchy-Schwarz inequalityu.vuv * u=(4,1,1)v=1,2,3u.v=4*1+1*2+1*3=9u.v=9u=16+1+1=18v=1+4+9=15u.v=18*15=16.43916.43 Cauchy-Schwarz inequality holds. * u=(1,2,1,2,3)v=0,1,1,5,-2u.v=0+2+1+10+-6=7u.v=7=7u=1+4+1+4+9=19v=0+1+25+4=31u.v=19*31=24.26724.26Cauchy-Schwarz inequality holdsQuestion 3(a)The given planes are: 1 2x-y+z-1=0 2 2x-y+z+1=0Here:a=2, b=-1, c=1, d1=-1, d2=1As the planes are parallel; therefore the distance between the plane is:d=d1-d2a2+b2+c2= -1-122+(-1)2+12=-24+1+1=26=630.816496distance between parrallel planes=d=0.816496Question 3(b)We know that if a and b are orthogonal vectors, then:a.b=0 * Here v.w=a,-b.(-b,a)=a*-b+b*a=-ab+ab=0Since v.w=0 v w are orthogonal vectors * Let another unit vector be:u=(cos,sin)u=cos2+sin2=1u.v=cos,sin.-3,4=1-3cos+4sin=04sin=3costan=34=tan-134=+tan-134when = tan-134u=(costan-134, sin (tan-134))=(45,35)when =+tan-134u=(cos+tan-134, sin (+tan-13 4))=(-45,-35)Question 4(a)x1+3x2-4x3=0x1+2x2+3x3=013-4123 x1x2x3=00The augmented matrix is13-412300Implementing row operations on the augmented matrixR2-R113-40-1700R1+3.R210170-1700-1.R2101701-700x1+17x3=0 x1=-17x3x2-7x3=0 x2=7x3Let x3=t x1=-17t x2=7t tRThe solution is:=-17t,7t,ttR }Which are orthogonal to:1,3,-4 (1,2,3)Question 4(b) * a=-3,2,-1b=(0,-2,-2)