Find the Null Space [[1,-2,2,-3],[2,-4,9,-2],[-3,6,-6,9]]

Problem

Null Space*[[1,−2,2,−3],[2,−4,9,−2],[−3,6,−6,9]]

Solution

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

A=[[1,−2,2,−3],[2,−4,9,−2],[−3,6,−6,9]]

2. Perform row operations to transform the matrix into row-echelon form. Add 3 times the first row to the third row, and subtract 2 times the first row from the second row.

[[1,−2,2,−3],[0,0,5,4],[0,0,0,0]]

3. Divide the second row by 5 to get a leading 1

[[1,−2,2,−3],[0,0,1,0.8],[0,0,0,0]]

4. Reduce to reduced row-echelon form (RREF) by subtracting 2 times the second row from the first row.

[[1,−2,0,−4.6],[0,0,1,0.8],[0,0,0,0]]

5. Identify free variables by looking at columns without pivots. Here, (x_2) and (x_4) are free variables. Let (x_2)=s() and (x_4)=t

(x_1)−2*s−4.6*t=0

(x_3)+0.8*t=0

6. Express the solution in vector form by solving for the pivot variables (x_1) and (x_3) in terms of s() and t

(x_1)=2*s+4.6*t

(x_3)=−0.8*t

7. Write the basis for the null space by separating the parameters s() and t

x=s*[[2],[1],[0],[0]]+t*[[4.6],[0],[−0.8],[1]]

Final Answer

Null Space=span*{[[2],[1],[0],[0]],[[4.6],[0],[−0.8],[1]]}

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