Determine if Linear

Problem

[[7],[2],[5]]*u+[[2],[−4],[8]]*v+[[5],[2],[8]]*w=[[35],[−18],[102]]

Solution

1. Write the vector equation as a system of linear equations.

7*u+2*v+5*w=35

2*u−4*v+2*w=−18

5*u+8*v+8*w=102

2. Set up the augmented matrix for the system.

[[7,2,5,35],[2,−4,2,−18],[5,8,8,102]]

3. Simplify the second row by dividing by 2

[[7,2,5,35],[1,−2,1,−9],[5,8,8,102]]

4. Swap the first and second rows to create a leading 1 in the first row.

[[1,−2,1,−9],[7,2,5,35],[5,8,8,102]]

5. Eliminate the first entries in the second and third rows using row operations (R_2)−7*(R_1)⇒(R_2) and (R_3)−5*(R_1)⇒(R_3)

[[1,−2,1,−9],[0,16,−2,98],[0,18,3,147]]

6. Simplify the third row by dividing by 3

[[1,−2,1,−9],[0,16,−2,98],[0,6,1,49]]

7. Eliminate the second entry in the second row using (R_2)−2*(R_3)⇒(R_2)

[[1,−2,1,−9],[0,4,−4,0],[0,6,1,49]]

8. Simplify the second row by dividing by 4

[[1,−2,1,−9],[0,1,−1,0],[0,6,1,49]]

9. Eliminate the second entry in the third row using (R_3)−6*(R_2)⇒(R_3)

[[1,−2,1,−9],[0,1,−1,0],[0,0,7,49]]

10. Solve for w using the third row.

7*w=49⇒w=7

11. Solve for v using the second row and back-substitution.

v−w=0⇒v−7=0⇒v=7

12. Solve for u using the first row and back-substitution.

u−2*v+w=−9⇒u−2*(7)+7=−9⇒u−7=−9⇒u=−2

Final Answer

[[7],[2],[5]]*(−2)+[[2],[−4],[8]]*(7)+[[5],[2],[8]]*(7)=[[35],[−18],[102]]

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