Find the Null Space [[5,8,17,-28],[5,2,-7,8],[3,5,11,-18]]

Problem

Null Space*[[5,8,17,−28],[5,2,−7,8],[3,5,11,−18]]

Solution

1. Set up the system by representing the null space as the set of all vectors x such that A*x=0

A=[[5,8,17,−28],[5,2,−7,8],[3,5,11,−18]]

2. Perform row operations to transform the matrix into row-echelon form. Subtract the first row from the second row ((R_2)−(R_1)→(R_2).

[[5,8,17,−28],[0,−6,−24,36],[3,5,11,−18]]

3. Simplify the second row by dividing by −6 (−1/6*(R_2)→(R_2).

[[5,8,17,−28],[0,1,4,−6],[3,5,11,−18]]

4. Eliminate the first column in the third row by performing (R_3)−3/5*(R_1)→(R_3)

[[5,8,17,−28],[0,1,4,−6],[0,0.2,0.8,−1.2]]

5. Eliminate the second column in the third row by performing (R_3)−0.2*(R_2)→(R_3)

[[5,8,17,−28],[0,1,4,−6],[0,0,0,0]]

6. Reduce to reduced row echelon form (RREF) by performing (R_1)−8*(R_2)→(R_1) and then dividing (R_1) by 5

[[1,0,−3,4],[0,1,4,−6],[0,0,0,0]]

7. Identify free variables (x_3) and (x_4) and express the pivot variables (x_1) and (x_2) in terms of them.

(x_1)=3*(x_3)−4*(x_4)

(x_2)=−4*(x_3)+6*(x_4)

8. Write the solution in vector form by factoring out the free variables.

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=(x_3)*[[3],[−4],[1],[0]]+(x_4)*[[−4],[6],[0],[1]]

Final Answer

Null Space=span*{[[3],[−4],[1],[0]],[[−4],[6],[0],[1]]}

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