Find the Null Space

Problem

null*[[6,−4,−10,−2],[−4,6,10,−2],[0,0,0,0],[2,2,0,−4]]

Solution

1. Set up the matrix equation A*x=0 to find the vectors x that satisfy the condition for the null space.

A=[[6,−4,−10,−2],[−4,6,10,−2],[0,0,0,0],[2,2,0,−4]]

2. Perform row operations to transform the matrix into reduced row echelon form (RREF). Divide the fourth row by 2 and swap it with the first row.

[[1,1,0,−2],[−4,6,10,−2],[6,−4,−10,−2],[0,0,0,0]]

3. Eliminate the entries in the first column below the pivot. Add 4 times the first row to the second, and subtract 6 times the first row from the third.

[[1,1,0,−2],[0,10,10,−10],[0,−10,−10,10],[0,0,0,0]]

4. Simplify the second row by dividing by 10 then use it to eliminate the entry in the third row.

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

5. Subtract the second row from the first row to reach the final RREF.

[[1,0,−1,−1],[0,1,1,−1],[0,0,0,0],[0,0,0,0]]

6. Identify the free variables. Here, (x_3) and (x_4) are free variables. Let (x_3)=s() and (x_4)=t

(x_1)−(x_3)−(x_4)=0⇒(x_1)=s+t

(x_2)+(x_3)−(x_4)=0⇒(x_2)=−s+t

7. Express the solution vector x in terms of the parameters s() and t

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

Final Answer

null*[[6,−4,−10,−2],[−4,6,10,−2],[0,0,0,0],[2,2,0,−4]]=span*{[[1],[−1],[1],[0]],[[1],[1],[0],[1]]}

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