Exercise 1.53. For which scalars a and b is aU + bV also a solution 1.55 For eac

Question

Exercise 1.53. For which scalars a and b is aU + bV also a solution 1.55 For each system: () Write the augmented matrix A. (i) Find all solutions (if any exist). Express your answer in parametric form and give the translation vector and the spanning vectors. State whether the solution is a line or plane or neither. (iii) If one of the rows of the augmented matrix becomes zero during the solution process, explicitly exhibit one row of A as a linear combination of the other rows. x-3y = 2 -2x + 6y =-4 x+ 3y+ z=1 2x+ 4y + 7z = 2 3x 10y52-7 x+ 3y + z=1 x+ 3y+ z=1 2x+ 4y + 7z 2 4x+10y + 9 7 4x+10y + 9z = 4 x +2t2+ 3 x-y +2z-2w = 1 2x+ y +3w = 4 2x+3y +2z =6 12+4x3 +3x4 2x3 +2x4 = 4

Explanation / Answer

(a)

a =

1 -3 2
-2 6 -4

R2=R2+2R1

a =

1 -3 2
0 0 0

second equation is obtained by multipling (-2) to first equation(one equation is linear combination of other)

so there is one independent equation of x and y

cant solve the system,it has infinite solutions

(b)

a=1 2 1 0 1
0 1 4 3 2
0 0 2 2 4

solving using Gauss-Jordan method,

a=

1.0000 0 0 0 -2.1538
0 1.0000 0 0 -1.1538
0 0 1.0000 0 -2.8462
0 0 0 1.0000 3.6923

x1= -2.1538

x2= -1.1538

x3=  -2.8462

x4= 3.6923

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