Find the Null Space

Problem

null*[[−3,6,−1,1,−7],[1,−2,2,3,−1],[2,−4,5,8,−4]]

Solution

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

[[−3,6,−1,1,−7,0],[1,−2,2,3,−1,0],[2,−4,5,8,−4,0]]

2. Swap rows to place a 1 in the top-left pivot position.

[[1,−2,2,3,−1,0],[−3,6,−1,1,−7,0],[2,−4,5,8,−4,0]]

3. Eliminate entries below the first pivot by performing (R_2)←(R_2)+3*(R_1) and (R_3)←(R_3)−2*(R_1)

[[1,−2,2,3,−1,0],[0,0,5,10,−10,0],[0,0,1,2,−2,0]]

4. Normalize the second row by performing (R_2)←1/5*(R_2)

[[1,−2,2,3,−1,0],[0,0,1,2,−2,0],[0,0,1,2,−2,0]]

5. Eliminate the entry below the second pivot by performing (R_3)←(R_3)−(R_2)

[[1,−2,2,3,−1,0],[0,0,1,2,−2,0],[0,0,0,0,0,0]]

6. Reduce to Reduced Row Echelon Form (RREF) by performing (R_1)←(R_1)−2*(R_2)

[[1,−2,0,−1,3,0],[0,0,1,2,−2,0],[0,0,0,0,0,0]]

7. Identify free variables by looking for columns without pivots. The pivots are in columns 1 and 3, so (x_2) (x_4) and (x_5) are free variables.

(x_1)=2*(x_2)+(x_4)−3*(x_5)

(x_3)=−2*(x_4)+2*(x_5)

8. Express the solution vector in terms of the free variables (x_2),(x_4),(x_5)

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=(x_2)*[[2],[1],[0],[0],[0]]+(x_4)*[[1],[0],[−2],[1],[0]]+(x_5)*[[−3],[0],[2],[0],[1]]

Final Answer

null(A)=span*{[[2],[1],[0],[0],[0]],[[1],[0],[−2],[1],[0]],[[−3],[0],[2],[0],[1]]}

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