Find the Null Space

Problem

Null Space*[[1,1,−2,1,5],[0,1,0,−1,−2],[0,0,8,0,16]]

Solution

1. Write the augmented matrix for the homogeneous system A*x=0 to find the vectors x such that:

[[1,1,−2,1,5],[0,1,0,−1,−2],[0,0,8,0,16]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[0],[0],[0]]

2. Simplify the third row by dividing by 8

[[1,1,−2,1,5],[0,1,0,−1,−2],[0,0,1,0,2]]

3. Perform row reduction to reach Reduced Row Echelon Form (RREF) by eliminating the entry above the pivot in the third column ((R_1)←(R_1)+2*(R_3):

[[1,1,0,1,9],[0,1,0,−1,−2],[0,0,1,0,2]]

4. Eliminate the entry above the pivot in the second column ((R_1)←(R_1)−(R_2):

[[1,0,0,2,11],[0,1,0,−1,−2],[0,0,1,0,2]]

5. Identify the pivot variables ((x_1),(x_2),(x_3) and the free variables ((x_4),(x_5). Write the system of equations:

(x_1)+2*(x_4)+11*(x_5)=0

(x_2)−(x_4)−2*(x_5)=0

(x_3)+2*(x_5)=0

6. Express the pivot variables in terms of the free variables (x_4)=s() and (x_5)=t

(x_1)=−2*s−11*t

(x_2)=s+2*t

(x_3)=−2*t

7. Write the solution vector in parametric form:

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=s*[[−2],[1],[0],[1],[0]]+t*[[−11],[2],[−2],[0],[1]]

Final Answer

Null Space=span*{[[−2],[1],[0],[1],[0]],[[−11],[2],[−2],[0],[1]]}

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